Precalculus Chapters 2, 4, 5, 6, 7, and Final Exam
For each quadrant on the cartesian coordinate plane, what is the value of x, y, r, and which functions are positive?
(1) For the first quadrant, x>0, y>0, r>0, and all functions are positive. (2) For the second quadrant, x<0, y>0, r>0, and the functions sine and cosecant are positive. (3) For the third quadrant, x<0, y<0, r>0, and the functions tangent and cotangent are positive. (4) For the fourth quadrant, x>0, y<0, r>0, and the functions cosine and secant are positive (pgs. 512, 514).
Good Angles
0°: 0 radians, (1,0) 30°: π/6 radians, (√(3)/2, 1/2) 45°: π/4 radians, (√(2)/2, √(2)/2) 60°: π/3 radians, (1/2, √(3)/2) 90°: π/2 radians, (0,1)
Quadrantal Angles
A angle in standard position is said to be quadrantal if and only if the terminal side of the angle is coincident with any one of the: (a) nonnegative horizontal axis, (b) nonnegative vertical axis, (c) nonpositive horizontal axis, or (d) nonpositive vertical axis (pgs. 501, 509, ICN).
Degree
A common unit for measuring angles. 360 degrees is equivalent to one full rotation; thus, 1° is equal to 1/360th of a full rotation (pg. 498).
Periodic Function
A function f such that f(x) = f(x + np), for every real number x in the domain of f, every integer n, and some positive real number p. The least possible positive value of p is the period, one full completion of a phenomenon, of the function (pg. 592).
Even Function
A function is even if and only if for all x-values in the domain of f, f(-x) = f(x) (pg. 654).
Odd Function
A function is odd if and only if for all x-values in the domain of f, f(-x) = -f(x) (pg. 654).
Function
A function must satisfy three requirements: (1) Start with a pair of not necessarily distinct nonempty sets usually designated as the first set and second set, (2) Each element of the first set is assigned a partner from the second set, and (3) No element from the first set is assigned two or more partners from the second set.
What is the definition of a function?
A function must satisfy three requirements: (1) start with a pair of not necessarily distinct nonempty sets usually designated the first set and the second set, (2) each element of the first set is assigned a partner from the second set, and (3) no element from the first set is assigned two or more partners from the second set.
When is a graph symmetrical about the horizontal axis?
A graph in the cartesian coordinate plane is symmetrical about the horizontal axis if and only if for every point (x,y) on the graph, (x,-y) is also a point on the graph.
When is a graph symmetrical about the origin (the point (0,0))?
A graph on the cartesian coordinate plane is symmetrical about the origin if and only if for every point (x, y) on the graph, (-x,-y) is also a point on the graph.
When is a graph symmetrical about the vertical axis?
A graph on the cartesian coordinate plane is symmetrical about the vertical axis if and only if for every point (x, y) on the graph, (-x,y) is also a point on the graph.
What is a horizontal intercept?
A horizontal intercept is a real-valued function f is a point on the graph of f that is also on the horizontal axis. Thus, horizontal intercepts take the form (?, 0).
Phase Shift
A horizontal translation of circular functions (pg. 605).
Exact Number
A number that represents the result of counting or that results from theoretical work and is not the result of a measurement (pg. 538).
Ray
A portion of a line that begins at some point on the line and extends indefinitely in one direction and includes all points on the line in that direction (pg. 498, ICN).
When does a function have a local maximum and what is it called?
A real-valued function f has a local maximum at a real number c if and only if c is in the domain of f and there exists an open interval I containing c such that for all real numbers x, if x is in I and x is in the domain of f and x is not equal to c, then f(x) ≤ f(c). The point (c, f(c)) is called a local maximum of f.
When does a function have a local minimum and what is it called?
A real-valued function f has a local minimum at a real number c if and only if c is in the domain of f and there exists an open interval I containing c such that for all real numbers x, if x is in I and x is in the domain of f and x is not equal to c, then f(x) ≥ f(c). The point (c, f(c)) is called a local minimum of f.
When is a function an even function?
A real-valued function f is an even function if and only if for all real numbers x in the domain of f, the following conditions are met: (1) -x is also in the domain of f, and (2) f(-x) = f(x).
When is a function an odd function?
A real-valued function f is an odd function if and only if for all real numbers x in the domain of f, the following conditions are met: (1) -x is also in the domain of f, and (2) f(-x) = -f(x).
When is a function increasing on interval I?
A real-valued function f is increasing on an interval I if and only if for every pair of distinct real numbers x1 and x2 in the interval I, if x1 < x2, then f(x1) < f(x2).
When is a function constant on the interval I?
A real-valued function is constant on the interval I if and only if for every pair of distinct real numbers x1 and x2 in the interval I, if x1 < x2 then f(x1) = f(x2).
When is a function decreasing on interval I?
A real-valued function is decreasing on an interval I if and only if for every pair of distinct real numbers x1 and x2 in the interval I, if x1 < x2, then f(x1) > f(x2).
Minute
A unit for measuring angles, a subdivision of degrees. 1 minute is 1/60th of a degree; thus, 1' is equal to 1/21,600th of a full rotation (ICN).
Second
A unit for measuring angles, a subdivision of degrees. 1 second is 1/60th of a minute and 1/3600th of a degree; thus, 1" is equal to 1/1,296,000th of a full rotation (ICN).
Grad
A unit for measuring angles. 400 grads is equivalent to one full rotation; thus, 1 grad is equal to 1/400th of a full rotation (ICN).
What is a vertical intercept?
A vertical intercept of a real-valued function f is a point on the graph of f that is also on the vertical axis. Thus, vertical intercepts take the form (0, ?).
Vertical Asymptote
A vertical line that the graph of a function approaches but does not intersect. As the x-values get closer and closer to the line, the function values increase or decrease without bound (pg. 616).
First Quadrant Angle
An angle in standard position is called a first quadrant angle if and only if all the interior points on its terminal side are in the first quadrant (ICN).
Fourth Quadrant Angle
An angle in standard position is called a fourth quadrant angle if and only if all the interior points on its terminal side are in the fourth quadrant (ICN).
Second Quadrant Angle
An angle in standard position is called a second quadrant angle if and only if all the interior points on its terminal side are in the second quadrant (ICN).
Third Quadrant Angle
An angle in standard position is called a third quadrant angle if and only if all the interior points on its terminal side are in the third quadrant (ICN).
Right Triangle Trigonometric Functions
The trigonometric functions can be defined by a right triangle (with an acute angle A in standard position) (pg. 521): (1) sin(A) = opposite/hypotenuse (2) cos(A) = adjacent/hypotenuse (3) tan(A) = opposite/adjacent (4) csc(A) = hypotenuse/opposite (5) sec(A) = hypotenuse/adjacent (6) cot(A) = adjacent/opposite
Argument
The variable/expression in the parenthesis of a function (pg. 605).
What is one technique to avoid sign errors when finding the trigonometric functions' values of an angle?
To avoid sign errors when finding the trigonometric functions' values of an angle, sketch it in standard position (pg. 527).
How does a person determine the half-way point and quarter points of a period?
To determine the half-way point in a period, add the x-values of the endpoints together and divide by 2. Repeat the process to find the quarter points (pg. 596).
How does a person determine the period length of y=sin(bx)?
To determine the period length of y=sin(bx), divide 2π by b (pg. 596).
How does one maintain accuracy?
To maintain accuracy, always use given information in your calculations as much as possible and avoid rounding during intermediate steps (pg. 540).
How do you solve for the values in an equilateral triangle?
To solve for the values of in an equilateral triangle (pg. 523): (1) Break the triangle into two equally sized right triangles, and (2) use the right triangle trigonometric functions to determine the length of the sides. The degrees for the top angle of the right triangles is always half of the whole angle for the equilateral, the bottom left corner angle is always 90°, and the final one is always the remainder of 180° - (90° + X) (pg. 523).
Diameter of the Circle
Twice the fixed distance r or a line segment from a point on a circle passing through the center of the circle to another point on the circle (ICN).
Complementary Angles
Two angles are complementary if and only if each angle has a positive degree measure and the sum of their degree measures is 90 degrees (pg. 499, ICN).
Supplementary Angles
Two angles are supplementary if and only if each angle has a positive degree measure and the sum of their degree measures is 180 degrees (pg. 499, ICN).
What determines the name of a line?
Two points A and B uniquely determine a line AB (pg. 498, ICN).
Angle
Two rays that share a common endpoint, or equivalently two rays that have the same vertex (pg. 498, ICN).
Double-Angle Identities
When A = B in the identities for the sum of two angles, double-angle identities result. The double-angle identities are (pgs. 683-684): (1) cos(2A) = cos²(A) - sin²(A) (2) cos(2A) = 2cos²(A) - 1 (3) cos(2A) = 1 - 2sin²(A) (4) sin(2A) = 2sin(A)cos(A) (5) tan(2A) = (2tan(A))/(1-tan²(A))
When a trigonometric equation is in quadratic form and it cannot be solved using the zero-factor property, what should be done?
When a trigonometric equation is in quadratic form, ax² + bx + c = 0, and it cannot be solved using the zero-factor property, use the quadratic equation (pg. 714).
Standard Position
An angle is said to be in standard position if and only if (pg. 501, ICN): (1) its vertex is placed at the origin of a Cartesian coordinate system, and (2) its initial side is coincident with the nonnegative axis.
Right Angle
An angle measuring 0.25 rotations or 90° (pg. 498, ICN).
Straight Angle
An angle measuring 0.5 rotations or 180°. Equivalently, a straight angle is one for which the union of the initial and terminal sides is a (straight) line (pg. 498, ICN).
Acute Angle
An angle measuring more than 0°and less than 90° (pg. 498, ICN).
Obtuse Angle
An angle measuring more than 90° but less than 180° (pg. 498, ICN).
Central Angle
An angle whose vertex is coincident with the center of a circle (ICN).
Radian
An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian (pg. 566).
How is an arc measured?
An arc is measured in a counterclockwise direction if its length is greater than zero starting from the point (1, 0) and it is is measured in a clockwise direction if its length is less than zero starting from the point (1,0) (pg. 578).
Identity
An equation that is satisfied by every value in the domain of its variable (pg. 654).
Identity
An identity is an equation that is true for all values of the variable for which both sides are defined (ICN).
What are angles often named after?
Angles are often named after their vertex or the points that make up the angle with the vertex's letter in the center (pg. 498).
Co-terminal Angles
Angles with the same initial and terminal sides, but different amounts of rotation. Their measures differ by measures of 360° (pg. 501, ICN).
Abcissa
Another name for a point's x-value (pg. 630).
Ordinate
Another name for a point's y-value (pg. 630).
Arc's Length Equation
Arcs, intercepted (drawn) on a circle with a radius r and a central angle measuring θ radians, have lengths equal to the product of the radius multiplied by the radian: s = rθ, where θ is in radians (pg. 569).
For each inverse function, what quadrants of the unit circle do their ranges inhabit?
Arcsine's range inhabits quadrants 1 and 4, arccosine's range inhabits quadrants 1 and 2, arctangent's range inhabits quadrants 1 and 4, arc-cosecant inhabits quadrants 1 and 4, arc-secant inhabits quadrants 1 and 2, and arc-cotangent inhabits quadrants 1 and 2 (pg. 703).
What unit of measure must calculators be in when they are used to find circular value functions?
Calculators must be in radian mode when they are used to find circular value functions (pg. 580).
What does "compound interest" mean and which banking formula belongs to it?
Compound interest means that the interest pays on the total amount in the account (previous interest and principle), and the banking formula that belongs to it is: A = P(1 + r/n)^(n * t).
What does "continuous compound interest" mean and which banking formula belongs to it?
Continuous compound interest means that the interest pays on the total amount in the account and never stops, and the banking formula that belongs to it is: A = P(e)^(r * t).
Does cos(A-B) equal cos(A) - cos(B)?
Cos(A-B) does not equal cos(A) - cos(B) (pg. 669).
Is cosine an even or odd function?
Cosine is an even function, i.e. cos(-θ) = cos(θ) (pg. 654).
What is cosine of a sum or difference?
Cosine of a sum or difference (pg. 670): cos(A+B) = cos(A)cos(B)-sin(A)sin(B), cos(A-B) = cos(A)cos(B)+sin(A)sin(B)
What equations must be used when graphing cotangent in a calculator?
Cotangent must be graphed with the equations 1/tan(x) or cos(x)/sin(x) in a calculator (pg. 618).
Inverse Cosine Function
Defined as arccos(x) = y, inverse cosine means that cos(y)=x for 0≤y≤π. We can think of these equations as "y is the number (angle) in the interval [0,π] whose cosine is x". You're basically swapping the domain and range (like all other inverses) (pg. 700).
Inverse Cosecant
Defined as arccosecant(x) = y, inverse cosecant means that csc(y)=x for -π/2 ≤ y ≤ π/2 (pg. 702).
Inverse Cotangent
Defined as arccot(x) = y, inverse cotangent means that cot(y) = x for 0 ≤ y ≤ π, y ≠ π/2 (pg. 702).
Inverse Secant
Defined as arcsec(x)=y, inverse secant means that sec(y)=x for 0 ≤ y ≤ π, y ≠ π/2 (pg. 702).
Inverse Sine Function
Defined as arcsin(x) = y, inverse sign means that sin(y) = x for -π/2≤y≤π/2. We can think of these equations as "y is the number (angle) in the interval [-π/2, π/2] whose sine is x". You're basically swapping the domain and range (like all other inverses). Its graph is symmetric with respect to the origin, so the function is odd. For all x in the domain, arcsin(-x) = -arcsin(x) (pgs. 698-699).
Inverse Tangent Function
Defined as arctan(x)=y, inverse tangent means that tan(y)=x for -π/2 < y < π/2. We can think of these equations as "y is the number (angle) in the interval (-π/2, π/2) whose tangent is x". Its graph is symmetric with respect to the origin, so the function is an odd function. The lines y=-π/2 and y=π/2 are horizontal asymptotes (pgs. 701-702).
Cosine Function
Defined as f(x) = cos(x), cosine is a periodic function with a domain of (-∞, ∞), a range of [-1. 1], x-intercepts that take the form (2n + 1)(π/2) (n is an integer), and periods that are 2π long. The cosine function is symmetrical about the y-axis (pg. 594).
Cotangent Function
Defined as f(x) = cot(x), cotangent is a periodic function that repeats every π, has a domain when x ≠ nπ, a range from (-∞, ∞), asymptotes that take the form nπ with n being an integer, and is symmetrical about the origin (pg. 618).
Cosecant Function
Defined as f(x) = csc(x), cosecant has periods every 2π, asymptotes at x = nπ where n is a whole number, x-values when x ≠ nπ where n is a whole number, a range from (-∞, -1]U[1,∞), and it is symmetrical about the origin (pg. 627).
Secant Function
Defined as f(x) = sec(x), secant is a periodic function that repeats every 2π, has a domain where x ≠ (2n + 1)(π/2) and n is an integer, a range from (-∞, -1]U[1,∞), asymptotes that take form (2n + 1)(π/2) where n is an integer, and it is an even function (pg. 626).
Sine Function
Defined as f(x) = sin(x), sine is a periodic function with a domain of (-∞, ∞), a range of [-1, 1] that corresponds to the y-values (sine values) of the unit circle, x-intercepts that take the form nπ (n is an integer), and periods that are 2π long. The sine function is symmetrical about the origin, so it is an odd function, f(-x) = -f(x) (pg. 593).
Tangent Function
Defined as f(x) = tan(x), it has a domain where x ≠ (2n + 1)(π/2) and n is an integer, a range from (-∞, ∞), vertical asymptotes at the values x = (2n + 1)(π/2), x-intercepts that take the form nπ where n is an integer, periods π long, and its graph is symmetrical about the origin (pg. 617).
Domain, Range, Period, Even/Odd, and Amplitude of Cosine
Domain: (-∞, ∞) Range: [-1, 1] Period: 2π Even/Odd: Even f(-x) = f(x) Amplitude: 1
Domain, Range, Period, Even/Odd, and Amplitude of Sine
Domain: (-∞, ∞) Range: [-1, 1] Period: 2π Even/Odd: Odd f(-x) = -f(x) Amplitude: 1
Domain, Range, Period, Even/Odd, and Amplitude of Tangent
Domain: A.R.N except π/2 + 2πn and except 3π/2 + 2πn (n is a whole number) Range: (-∞, ∞) Period: π Even/Odd: Odd f(-x) = -f(x) Amplitude: None
Domain, Range, Period, Even/Odd, and Amplitude of Cosecant
Domain: A.R.N. except π + 2πn Range: (-∞,-1] U [1,∞) Period: 2π Even/Odd: Odd f(-x) = -f(x) Amplitude: None
Domain, Range, Period, Even/Odd, and Amplitude of Secant
Domain: A.R.N. except π/2 + 2πn and except 3π/2 + 2πn Range: (-∞,-1] U [1,∞) Period: 2π Even/Odd: Odd f(-x) = -f(x) Amplitude: None
Domain, Range, Period, Even/Odd, and Amplitude of Cotangent
Domain: A.R.N. except πn (n is a whole number) Range: (-∞, ∞) Period: π Even/Odd: Odd f(-x) = -f(x) Amplitude: None
Conditional Equations
Equations, such as y = r/x, which are satisfied by some values but not others (pg. 712).
What is the rule when solving a trigonometric equation?
When solving a trigonometric equation, avoid dividing by a variable function, such as sin(θ) (pg. 713).
What should you be aware of when taking the square root of a number in a trigonometric problem?
When taking the square root of a number in a trigonometric problem, be sure to choose the sign +, -, or ± based on the quadrant of θ and the function being evaluated (pg. 656).
What equations must be used when graphing cosecant and secant functions in a calculator?
You must graph the reciprocal identities of cosecant and secant, i.e. 1/sin(x) and 1/cos(x) when graphing them in a calculator (pg. 628).
What should you round the final answer in a problem to?
You should round the final answer to the same number of significant digits as the number in a calculation with the least amount of significant digits, excluding constants and exact numbers, which have infinitely many significant digits (pg. 539).
What is arc-cosecant evaluated as in the calculator?
arccsc(x) is evaluated as arcsin(1/x) when using a calculator (pg. 704).
What is arc-secant evaluated as in the calculator?
arcsec(x) is evaluated as arccos(1/x) when using a calculator (pg. 704).
Co-functional Identities
For an acute angle A, the following identities hold because the sum of the two acute angles in a right triangle equal 90° and the total degrees in a right triangle equals 180° (pg. 522): (1) sin(A) = cos(90° - A), (2) cos(A) = sin(90° - A), (3) tan(A) = cot(90° - A), (4) csc(A) = sec(90° - A), (5) sec(A) = csc(90° - A), (6) cot(A) = tan(90° - A)
Reference Angle
For every non-quadrantal angle θ in standard position, the reference angle θ', is the acute angle made by the terminal side of the angle θ and the x-axis. The reference angle is always between 0° and 90° (pg. 524).
Angle of Elevation
For right triangles, the angle from point X to point Y (above X) is the acute angle formed by ray XY and a horizontal ray with its endpoint at X (pg. 541).
Angle of Depression
For right triangles, the angle from point X to point Y (below X) is the acute angle formed by ray XY and a horizontal ray with endpoint X (pg. 541).
What is function notation?
Function notation, f(x), has this meaning: if x is in the domain of f, then f(x) is the partner that function f assigns to x. If x is not in the domain of f, then f(x) does not have a meaning.
What is the composition of functions f and g?
Given a pair of functions f and g, the composition of f with g, denoted as f ∘ g, is the function defined by (f ∘ g)(x) = f(g(x)) for each x in the domain of g for which g(x) in the domain of f.
What is the difference of functions f and g?
Given that f and g are real valued functions, the difference of the functions f and g, denoted as f-g, is the function defined by (f-g)(x) = f(x) - g(x) for each x in the domains of both f and g.
What is the product of functions f and g?
Given that f and g are real valued functions, the product of the functions f and g, denoted as fg, is the function defined by (fg)(x) = f(x)g(x) for each x in the domains of both f and g.
What is the quotient of functions f and g?
Given that f and g are real valued functions, the quotient of functions f and g, denoted (f/g) is the function defined by (f/g)(x) = (f(x))/(g(x)) for each x in the domains of both f and g where the value g(x) is not zero.
What is the sum of functions f and g?
Given that f and g are real-valued functions, the sum of the functions f and g, denoted as f+g, is the function defined by (f+g)(x) = f(x) + g(x) for each x in the domains of both f and g.
Line Segment
Given two points A and B, the line segment from A to B, denoted by AB, is A and B together with the portion of line AB between them (pg. 498, ICN).
What are the guidelines for sketching cosecant and secant functions of the form y=acsc(bx) or y = asec(bx) with b>0?
Guidelines for sketching cosecant and secant functions of the form y=acsc(bx) or y = asec(bx) with b>0 (pg. 628): (1) Graph the corresponding reciprocal function as a guide, using a dashed curve. For y=acsc(bx), use y=asin(bx) as a guide. For y=asec(bx), use y=acos(bx). (2) Sketch the vertical asymptotes. They will have equations of the form x=k, where k corresponds to an x-intercept of the graph of the guide function. (3) Sketch the graph of the desired function by drawing the typical u-shaped branches between the adjacent asymptotes. The branches will be above the graph of the guide function when the guide function values are positive and below the guide function values are negative.
What are the guidelines for sketching graphs of tangent and cotangent functions that take the form y = atan(bx) or y = acot(bx) with b>0?
Guidelines for sketching graphs of tangent and cotangent functions that take the form y = atan(bx) or y = acot(bx) with b>0 (pg. 619): (1) Determine the period π/b. To locate two adjacent vertical asymptotes, solve the following equation for x: For y = atan(bx): bx = -π/2 and bx = π/2 For y = acot(bx): bx = 0 and bx = π (2) Sketch the two vertical asymptotes found in Step 1. (3) Divide the interval formed by vertical asymptotes into four equal parts. (4) Evaluate the function for the first-quarter point, midpoint, and third-quarter point using the x-values found in Step 3. (5) Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods on the graph as necessary.
What are the guidelines for sketching graphs of the sine and cosine functions?
Guidelines for sketching graphs of the sine and cosine functions (pg. 598): (1) Find the period, 2π/b, start 0 on the x-axis, and lay off a distance of 2π/b. (2) Divide the interval into four equal parts. (3) Evaluate the function for each of the five x-values resulting from step 2. The points will be the maximum points, the minimum points, and the x-intercepts. (4) Plot the points found in step 3, and join them with a sinusoidal curve having amplitude |a|. (5) Draw the graph over additional periods as needed.
Amplitude
Half of the distance between the maximum and minimum y-values, the "height" of the graph starting from the x-axis (pg. 594).
What can having a quadrantal angle lead to?
Having a quadrantal angle can lead to some of the trigonometric functions being undefined because you cannot divide by zero (pg. 509).
What are some hints for verifying identities?
Hints for verifying identities (pg. 661): (1) Learn the fundamental identities and be aware of their equivalent forms. (2) Try to rewrite the more complicated side of the equation so that it is identical to the simpler side. (3) It is sometimes helpful to express all trigonometric functions in the equation in terms of sine and cosine and simplify the result. (4) Usually, any factoring or indicated algebraic operations (addition, subtraction, etc.) should be performed. (5) When selecting substitutions, keep in mind the side that is not changing, because it represents the goal. (5) If an expression contains 1 + sin(x), multiplying both the numerator and denominator by 1 + sin(x) would give 1 + sin²(x), which could be replaced with cos²(x). Similar procedures apply for 1-sin(x), 1+cos(x), and 1-cos(x).
When does a function have partner sharing?
If A and B are nonempty sets, a function f from A to B has partner sharing if and only if there is an element b of B such that there exists distinct element a and c of set A so that both f(a) = b and F(c) = b.
When is a function one to one?
If A and B are nonempty sets, a function f from A to B is one to one if and only if for all a in A and for all b in A, if f(a) = f(b), then a = b and for all a in A and for all b in A, if a ≠ b, then f(a) ≠ f(b).
If both sides of an identity appear equally complex, what should be done?
If both sides of an identity appear equally complex, work on the left and right side independently until each side is changed into some common third result. Each step on each side must be reversible (pg. 663).
What is the secant line of a function?
If c and d are distinct numbers in the domain of a real-valued function f, then the (unique) line through the points (c, f(c)) and (d, f(d)) is called the secant line of f from c to d.
What is the average rate of change of a function?
If c and d are distinct numbers in the domain of a real-valued function f, then the average rate of change of f from c to d is (f(d) - f(c)) / (d-c).
If no unit of angle measure is specified, what should a person assume it is?
If no unit of angle measure is specified, then the angle is understood to be measured in radians (pg. 568).
If the terminal side of a quadrantal angle lies along the x-axis, which functions are undefined?
If the terminal side of a quadrantal angle lies along the x-axis, the cotangent and cosecant functions are undefined (pg. 510).
If the terminal side of a quadrantal angle lies along the y-axis, which functions are undefined?
If the terminal side of a quadrantal angle lies along the y-axis, the tangent and secant functions are undefined (pg. 510).
What is arc-cotangent evaluated as in the calculator?
If x>0, then arccot(x) is evaluated as arctan(1/x) and if x<0, then arccot(x) is evaluated as arctan(1/x) + 180° (pg. 704).
What do the terms "annually", "semi-annually", "quarterly", "monthly", "bimonthly", "weekly", and "daily" mean in banking?
In banking, annually means interest is compounded once a year (n = 1), semi-annually means twice a year (n = 2), quarterly means four times a year (n = 4), monthly means twelve times a year (n = 12), bimonthly means twenty-four times a year (n = 24), weekly means fifty-two times a year (n = 52), and daily means 365 times a year (n = 365).
How does b change the form of y=sin(x) or y=cos(x)?
In general, the graph of a function of the form y=sin(bx) or y=cos(bx) for b>0 will have a period different from 2π when b≠1 (pg. 596).
If working with one side does not appear to be working, what should be done?
In practice, if working with one side does not seem to be effective, switch to the other side (pg. 664).
What is unique about angles and their reference angles in quadrant 1?
In quadrant 1, θ and θ' are the same (pg. 524).
Half-Angle Identities
In the following half-angle identities, the ± symbol indicates that the sign is chosen based on the function under consideration and the quadrant of A/2 (pg. 690): cos(A/2)=±√((1 + cos(A))/2) sin(A/2)=±√((1 - cos(A))/2) tan(A/2)=±√((1 - cos(A))/(1 + cos(A))) tan(A/2)=(sin(A))/(1+cos(A)) tan(A/2)=(1-cos(A))/(sin(A))
What does θ represent in trigonometric identities?
In trigonometric identities, θ can represent an angle in degrees, radians, or a real number (pg. 654).
What is point p and what does it determine?
Let point p (x, y) be any point, other than the origin, on the terminal side of an angle θ in standard position. The distance from the point p to the origin, the length of the terminal side, is r = √(x² + y²) (pgs. 506-507).
What are the six circular trigonometric functions of θ?
Let θ be a measure of an angle in standard position. Let P with coordinates (x, y) be the point of intersection of the terminal side of the angle with the unit circle x² + y² = 1. The six circular trigonometric functions of θ are therefore (pgs. 506-507, ICN): (1) sin(θ) = y/r (2) cos(θ) = x/r (3) tan(θ) = y/x (x ≠ 0) (4) csc(θ) = r/y (y ≠ 0) (5) sec(θ) = r/x (x ≠ 0) (6) cot(θ) = x/y (y ≠ 0)
Circular Trigonometric Functions
Let θ be a measure of an angle in standard position. Let P with coordinates (x, y) be the point of intersection of the terminal side of the angle with the unit circle x² + y² = 1. Then, define the (circular) trigonometric functions of θ by: (1) cos(θ) = x, (2) sin(θ) = y, (3) tan(θ) = y/x (x ≠ 0), (4) sec(θ) = 1/x (x ≠ 0), (5) csc(θ) = 1/y (y ≠ 0), and (6) cot(θ) = x/y (y ≠ 0).
(Circular) Trigonometric Functions (in terms of the unit circle)
Let θ be a measure of an angle in standard position. Let P with coordinates (x,y) be the point of intersection of the terminal side of this angle with the unit circle x² + y² = 1. Then define the (circular) trigonometric functions of θ by: cosθ = x/r, secθ = 1/x (x ≠ 0), sinθ = y/r, cscθ = 1/y (y ≠ 0), tanθ = y/x (x ≠ 0), cotθ = x/y (y ≠ 0)
What is method 1 for graphing y = c + asin[b(x-a)] or y = c + acos[b(x-d)] with b>0?
Method 1 for graphing y = c + asin[b(x-a)] or y = c + acos[b(x-d)] with b>0 (pg. 609): (1) Find an interval whose length is one period 2π/b by solving the three-party inequality 0≤b(x-d)≤2π. (2) Divide the interval into four equal parts to obtain five key x-values. (3) Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and points that intersect the line y = c, (4) Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|. (5) Draw the graph over additional periods as needed.
What is method 2 for graphing y = c + asin[b(x-a)] or y = c + acos[b(x-d)] with b>0?
Method 2 for graphing y = c + asin[b(x-a)] or y = c + acos[b(x-d)] with b>0 (pg. 609): (1) Graph y = asin(bx) or y = acos(bx). The amplitude of the function is |a| and the period is 2π/b. (2) Use translations to graph the desired function. The vertical transition is c units up if c>0 and |c| units down if c<0. The horizontal translation (phase shift) is d units to the right if d>0 and |d| units to the left if d<0.
How does one convert from degrees to radians?
Multiply a degree measure by π/180° and simplify to convert from degrees to radians (pg. 567).
How does one convert from radians to degrees?
Multiply a radian measure by 180°/π and simplify to convert from radians to degrees (pg. 567).
Side of an Angle
One of the two rays or line segments with a common endpoint that make up an angle (pg. 498, ICN).
Radian Measure
Place the vertex of an angle at the center of a circle of radius r. Let s be the arc length of the circle subtended by this angle. The radian measure θ of this angle is given by θ = s/r, provided that r and s are measured in the same linear units.
Radian Measure
Place the vertex of an angle at the center of a circle with radius r. Let s be the arc length of the circle subtended by the angle. The radian measure θ of the angle is given by θ = s/r provided that r and s are measured in the same linear units (ICN).
What does "simple interest" mean and which banking formula belongs to it?
Simple interest means that the interest pays only on the original investment in the account, and the banking formula that applies to it is: A = P + Prt.
Is sine an even or odd function?
Sine is an odd function, i.e. sin(-θ) = -sin(θ) (pg. 654).
What is sine of a sum or difference?
Sine of a sum or difference (pg. 673): sin(A+B) = sin(A)cos(B) + cos(A)sin(B) sin(A-B) = sin(A)cos(B) - cos(A)sin(B)
What are the steps for solving a trigonometric equation?
Steps for solving a trigonometric equation (pg. 715): (1) Decide whether the equation is linear or quadratic in form in order to determine the solution method. (2) If only one trigonometric function is present, solve the equation for that function. (3) If more than one trigonometric function is present, rewrite the equation so that one side equals 0. Then try to factor and apply the zero-factor property. (4) If the equation is quadratic in form, but not factorable, use the quadratic formula. Check that the solutions are in the desired interval. (5) Try using identities to change the form of the equation. It may be helpful to square each side of the equation first. In this case, check for extraneous solutions.
Is tangent an even or odd function?
Tangent is an odd function, i.e. tan(-θ) = tan(θ) (pg. 654).
What is tangent of a sum or difference?
Tangent of a sum or difference (pg. 673): (1) tan(A+B) = (tan(A) + tan(B))/(1 -tan(A)tan(B)) (2) tan(A-B) = (tan(A) - tan(B))/(1 + tan(A)tan(b)
What are the Functional values of 30°, 45°, and 60° for cos(θ)?
The Functional values of 30°, 45°, and 60° for cos(θ) are √(3)/2, √(2)/2, and 1/2 respectively (pg. 524).
What are the Functional values of 30°, 45°, and 60° for cot(θ)?
The Functional values of 30°, 45°, and 60° for cot(θ) are √(3), 1, and √(3)/3 respectively (pg. 524).
What are the Functional values of 30°, 45°, and 60° for csc(θ)?
The Functional values of 30°, 45°, and 60° for csc(θ) are 2, √(2), and (2√(3))/3 respectively (pg. 524).
What are the Functional values of 30°, 45°, and 60° for sec(θ)?
The Functional values of 30°, 45°, and 60° for sec(θ) are (2√(3))/3, √(2), and 2 respectively (pg. 524).
What are the Functional values of 30°, 45°, and 60° for sine(θ)?
The Functional values of 30°, 45°, and 60° for sine(θ) are 1/2, √(2)/2, and √(3)/2 respectively (pg. 524).
What are the Functional values of 30°, 45°, and 60° for tan(θ)?
The Functional values of 30°, 45°, and 60° for tan(θ) are √(3)/3, 1, and √(3) respectively (pg. 524).
What are the Pythagorean identities?
The Pythagorean identities are (pg. 655): (1) sin²(θ) + cos²(θ) = 1, (2) tan²(θ) + 1 = sec²(θ), and (3) 1 + cot²(θ) = csc²(θ).
Area of a Sector Equation
The area A of a sector from a circle with radius r and a central angle θ is equal to: A = (1/2)r²θ, where θ is in radians (pg. 572).
What unit does the calculator returns the results for arcsine(y/r), arccosine(x/r), and arctangent(y/x) in?
The calculator returns the results for arcsine(y/r), arccosine(x/r), and arctangent(y/x) in degrees (pg. 529).
Unit Circle
The circle with its center at the origin on the cartesian coordinate plane and a 1 unit radius; thus, the equation for the circumference of the unit circle is x² + y² = 1 (pg. 578).
Composition
The composition of function F with function G, denoted F ∘ G, is the function that is defined by (F ∘ G)(x) = F(G(x)).
What cue distinguishes a counterclockwise from a clockwise angle measurement?
The cue that distinguishes a counterclockwise from a clockwise angle measurement is that a counterclockwise measurement generates a positive number, while a clockwise measurement generates a negative number (pg. 498, ICN).
Quotient Identities
The different relationships between the trigonometric functions that are derived by dividing the functions together (pg. 516): (1) (sin(θ)/cos(θ)) = tan(θ) and (2) (cos(θ)/sin(θ)) = cot(θ).
Pythagorean Identities
The different relationships between the trigonometric functions that are derived from pythagorean's function. For all angles θ for which the function values are defined, these identities hold (pgs. 515, 516): (1) sin²(θ) + cos²(θ) = 1, (2) tan²(θ) + 1 = sec²(θ), and (3) cot²(θ) + 1 = csc²(θ).
Significant Digits
The digits in a number that are warranted being labeled accurate based on the precision of the measuring instrument (pg. 538).
Circumference
The distance around the circle, equal to 360°, 2πr, and (x-h)² + (y-k)² = r² (pg. 566, ICN).
What is the domain of quotient of f and g in set notation?
The domain of quotient of f and g is given in set notation as {x | x is the domain of f and x is the domain of g and g(x) ≠ 0}.
What is the domain of composition of f with g in set notation?
The domain of the composition of f with g, i.e. f ∘ g, in set notation is {x | x is in the domain of g and g(x) is the domain of f}.
What are the limits on the cotangent and cosecant functions' domain and why are there limits?
The domain of the cotangent and cosecant functions in symbolic form are {s | s ≠ nπ, where n is any integer}. There domains are limited to avoid divide-by-zero errors (pg. 580).
What are the limits on the sine and cosine functions' domain?
The domain of the sine and cosine functions in symbolic form are {s | s = (-∞, ∞)} (pg. 580).
What are the limits on the tangent and secant functions' domain and why are there limits?
The domain of the tangent and secant functions in symbolic form are {s | s ≠ (2n + 1)(π/2), where n is any integer}. These domains are limited to avoid divide-by-zero errors (pg. 580).
Why are the domains of the inverse sine, cosine, tangent, cosecant, secant, and cotangent functions restricted?
The domains of the inverse sine, cosine, tangent, cosecant, secant, and cotangent functions are all restricted to allow the functions to be one-to-one (otherwise they wouldn't be a function).
What are the domains of the sum, difference, and product of f and g?
The domains of the sum, difference, and product of f and g are those elements in both the domain of f and the domain of g. In set notation: {x | x is the domain of f and x is the domain of g}.
What are the equations for reference angles based on the terminal side's quadrant?
The equations for reference angles based on the terminal side's quadrant are (pg. 526): Q1: θ' = θ, Q2: θ' = 180° - θ, Q3: θ' = θ - 180°, and Q4: θ' = 360° - θ.
What are the even-odd identities?
The even-odd identities are (pg. 655): (1) sin(-θ) = -sin(θ), (2) cos(-θ) = cos(θ), (3) tan(-θ) = -tan(θ), (4) csc(-θ) = -csc(θ), (5) sec(-θ) = sec(θ), and (6) cot(-θ) = -cot(θ).
Radius of the Circle
The fixed distance r or a line segment from the center of the circle to a point on the circle (ICN).
Co-functional Identities
The following co-functional identities hold for any angle θ for which the functions are defined. The same identities can be obtained for a real number domain by replacing 90° with π/2 (pg. 672): (1) cos(90°-θ) = sin(θ), (2) sin(90°-θ) = cos(θ), (3) tan(90°-θ) = cot(θ), (4) sec(90°-θ) = csc(θ), (5) csc(90°-θ) = sec(θ), and (6) cot(90°-θ) = tan(θ).
What is the difference betwen y=asin(x) and y=sin(x) along with y=acos(x) and y=cos(x)?
The graph of y=asin(x) will have the same shape as y=sin(x) and the graph of y=acos(x) will have the same shape as y=cos(x) where a≠0, except their ranges will be [-|a|, |a|] and their amplitude will be |a| (pg. 595).
What is often used to name/mark an angle?
The greek letter theta, θ, is often used to name/mark an angle (pg. 499).
Periodic Functions of the Unit Circle
The periodic functions of the unit circle are (pg. 592): (1) sin(x) = sin(x + (n * 2π)), (2) cos(x) = cos(x + (n * 2π)), (3) f(x) = tan(x), (4) f(x) = cot(x), (5) f(x) = sec(x), and (6) f(x) = csc(x).
Is the procedure for verifying identities the same s that for solving equations?
The procedure for verifying identities is not the same as that for solving equations. Techniques used in solving equations, such as adding the same term to both sides, should not be used when working identities (pg. 660).
Product-to-Sum Identities
The product-to-sum identities are (pg. 687): cosAcosB = (1/2)[cos(A+B)+cos(A-B)] sinAsinB = (1/2)[cos(A-B)-cos(A+B)] sinAcosB = (1/2)[sin(A+B)+sin(A-B)] cosAsinB = (1/2)[sin(A+B)-sin(A-B)]
What is an alternative way to write the pythagorean identity?
The pythagorean identity, also the unit circle equation, can be written (pg. 579): cos²(x) + sin²(s) = 1.
What are the quotient identities?
The quotient identities are (pg. 655): (1) tan(θ) = (sin(θ)/cos(θ)), and (2) cot(θ) = (cos(θ)/sin(θ)).
What is the difference between the radian measure of an angle and the other forms of measurement?
The radian measure is a unitless measure of an angle in contrast to the other measures (degrees, rotations, grads) that have units. In other words, the term radian (or radians) is a label of convenience to be applied (or not) to a real number - and is typically applied when the real number is measuring an angle and generally not applied when an angle measure is not part of the context (ICN).
What are the ranges for the different trigonometric functions?
The ranges for the different trigonometric functions (pgs. 513, 514): sin(θ), cos(θ) = [-1, 1] = |y| ≤ 1 tan(θ), cot(θ) = (-∞, ∞) = A.R.N. sec(θ), csc(θ) = (-∞, 1]U[1,∞) = |y| ≥ 1
Why can you use any point p on the terminal side of an angle θ in standard position?
The reason you can use any point on the terminal side when calculating a trigonometric function is because the ratios of the different points p's values are the same (i.e. (10/5) = (20/10) = 2)) (pg. 508).
What are the reciprocal identities?
The reciprocal identities are (pg. 655): (1) cot(θ) = (1/tan(θ)), (2) sec(θ) = (1/cos(θ)), and (3) csc(θ) = (1/sin(θ)).
Reciprocal Identities
The reciprocal of a number, x = (1/x), that is true for all values of the angle θ for which all the expressions are defined and for which both functions are defined (pg. 511): (1) sin(θ) = 1/csc(θ) (2) cos(θ) = 1/sec(θ) (3) tan(θ) = 1/cot(θ) (4) csc(θ) = 1/sin(θ) (5) sec(θ) = 1/cos(θ) (6) cot(θ) = 1/tan(θ).
What is the reciprocal form of a trigonometric function equivalent to?
The reciprocal trigonometric function is equivalent to (fun(θ))^(-1) not fun^(-1)(θ) (pg. 511).
Fundamental Identities
The reciprocal, quotient, Pythagorean, and even-odd identities (pg. 654).
What is the reverse of function f?
The reverse of a function f, denoted Rf, is the relation from the range of f to the domain of f consisting of all reversed partnerships of f. Symbolically, Rf = {(b,a) | b is in the range of f, a is in the domain of f, and f(a) = b}.
What are the rules for significant digits?
The rules for significant digits (pg. 538): (1) all non-zero digits are significant, (2) all zero digits between non-zero digits are significant, and (3) all zero digits that are behind a non-zero digit and the decimal place are significant.
Circle
The set of points a fixed specified distance r from a given point C (ICN).
Reference Arc
The shortest arc from a point on a circle to the x-axis (pg. 578).
Initial Side
The side of an angle that the angle measurement begins on (pg. 498).
Terminal Side
The side of an angle that the angle measurement ends on (pg. 498).
What determines the sign of a reciprocal value?
The sign of a function value automatically determines the sign of the reciprocal function value because numbers that are reciprocals always have the same sign, i.e. -2/1 and 1/-2 (pg. 512).
Sector of a Circle
The space bounded by the rays of a central angle and the circle (pg. 571, ICN):
What are the standard points for the graph of cosecant?
The standard points for the graph of cosecant {(0,U), (π/6,2), (π/3,2√3/3), (π/2,1), (2π/3,2√3/3), (π,U), (3π/2,-1), and (2π,U)} (pg. 627).
What are the standard points for the graph of f(x) = tan(x)?
The standard points for the graph of f(x) = tan(x) are {(-π/2,U), (-π/4,-1), (0,0), (π/4,1), and (π/2,U)} (pg. 617).
What are the standard points for the graph of f(x)=cot(x)?
The standard points for the graph of f(x)=cot(x) are {(0,U), (π/4,1), (π/2,0), (3π/4,-1), and (π,U)} (pg. 618).
What are the standard points for the graph of secant?
The standard points for the graph of secant are {(-π/2,U), (-π/4,√2), (0,1), (π/4,√2), (π/2,U), (3π/4,-√2), (π,-1), (3π/2,U)} (pg. 626).
What are the standard points for y = cos(x)?
The standard points for y = cos(x) are {(0,1), (π/2,0), (π,-1), (3π/2,0), and (2π,1)} (pg. 595).
What are the standard points for y = sin(x)?
The standard points for y = sin(x) are {(0,0), (π/2,1), (π,0), (3π/2,-1), and (2π,0)} (pg. 595).
What are the standard points on the graph of inverse cosine?
The standard points on the graph of arccosine are {(-1,π), (-√2/2,3π/4), (0,π/2), (√2/2,π/4), (1,0)} (pg. 701).
What are the standard points on the graph of arcsine?
The standard points on the graph of arcsine are {(-1,-π/2), (-√2/2,-π/4), (0,0), (-√2/2,π/4), (1,π/2)} (pg. 699).
Vertex of the Ray
The starting (or ending point) of a ray (pg. 498, ICN).
What are the steps for finding trigonometric function values for any non-quadrantal angle θ?
The steps for finding trigonometric function values for any non-quadrantal angle θ are (pg. 527): (1) If θ>360° or if θ>0°, then find a co-terminal angle by adding or subtracting 360° as many times as needed to obtain a number greater than 0° but less than 360°. (2) Find the reference angle θ'. (3) Find the trigonometric functions' values for reference angle θ'. (4) Determine the correct signs for the values found in step 3. This gives the values of the trigonometric functions for angle θ.
What are the steps for solving applied trigonometric problems (word problems)?
The steps for solving applied trigonometric problems (word problems) are (pg. 541): (1) Draw a sketch and label it with the given information. Label the quantity to be found with a variable. (2) Use the sketch to write an equation relating the given quantities to the variable. (3) Solve the equation, and check that the answer makes sense.
Sum-to-Product Identities
The sum-to-product identities are (pg. 688): sinA + sinB = 2sin((A+B)/2)cos((A-B)/2) sinA - sinB = 2cos((A+B)/2)sin((A-B)/2) cosA+cosB = 2cos((A+B)/2)cos((A-B)/2) cosA-cosB = -2sin((A+B)/2)sin((A-B)/2)
