Precalculus Chapter 5

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).

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).

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).

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).

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

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 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).

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).

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).

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).

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).

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).

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 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).

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 unique about angles and their reference angles in quadrant 1?

In quadrant 1, θ and θ' are the same (pg. 524).

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)

Side of an Angle

One of the two rays or line segments with a common endpoint that make up an angle (pg. 498, ICN).

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 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).

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).

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 is often used to name/mark an angle?

The greek letter theta, θ, is often used to name/mark an angle (pg. 499).

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).

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).

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.

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).

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.