Algebra II : Chapter 12 Exam
Define *Angle of Elevation*
- always measured from *ground up* - it's an upward angle from a horizontal line
The _____________________________ of the graph of a sine or cosine function equals . . .
- amplitude - half the difference between the maximum and minimum values of the function
Define *Angle of Depression*
- angle formed by the horizontal and the observer's line of sight to an object below
What does cosθ = _______ and what does sin θ = ____?
- cosθ = x - sin θ = y
How do you find a reference angle?
- if the reference angle is in the first quadrant then, *θ' = θ* - if in any other quadrant, subtract the entire angle from the nearest quadrant angle to find the reference ex : 200° is close to the quadrant angle of 180°, so *200° - 180°* equals 20°, which is the reference angle
A ___________________________ has y-values that repeat at regular intervals. One complete pattern is called a ____________________, and the horizontal length is called a _________________________.
- periodic function - cycle - period
Cosecant, secant, and cotangent are known as ___________________________________ to the normal trigonometric functions.
- reciprocal functions
What does it mean to be a matching pair?
A matching pair has a side and an opposite angle that both have their appropriate numerical measurements.
Define *central angle*
An angle formed with the vertex being at the center of a circle and the sides are radii of the circle.
How do you turn degrees to radians?
multiply the number of degrees by π radians ----------- 180° ex: -30*°* / 1 • π radians / 180*°* = -π / 6
How do you turn radians to degrees
multiply the number of radians by 180° ----------- π radians ex: 5*π* / 2 • 180° / *π* = 450°
What is the equation for finding arc length?
n° ----- • 2πr 360°
How do you find *f*requency?
period = 1/f f = 1/period *Note* : f = cycles/second or Hertz
What is the formula for finding the radius of a circle?
r = ½d r = d/2
To find the radius or hypotenuse of a right triangle formed by the terminal side of an angle on a coordinate plane, you use the equation _______________________.
r = √x² + y²
The midline and the vertical shift have the ___________________ value.
same
What are the trigonometric functions that are used to find angle values greater than 90° or less than 0°
sin θ = *y*/r cos θ = *x*/r tan θ = *y/x*, x ≠ 0 csc θ = r/y, y ≠ 0 sec θ = r/x, x ≠ 0 cot θ = x/y, y ≠ 0 *r = 1* *Note* : These equations are difficult to use when the terminal side is in the quadrant, use SOH CAH TOA instead. If it is on the axis, use these formulas.
What are the equations for graphing trigonometric functions?
sine : *y = a sin bθ* cosine : *y = a cos bθ* tangent : *y = a tan bθ* amplitude : *|a|* period : *360 ( 2π ) / |b|* *NOTE* : Tangent does NOT have an amplitude, but it has a period, which is found by using the equation, *180°/ |b|*
When you solve for a triangle, you . . .
solve for all of the sides and angles
Define amplitude
the distance from the midline to the highest point or to the lowest point
If two legs are congruent, then . . .
their opposite base angles are congruent
What is the equation for the translation of any trigonometric function?
y = *a* [ sin, cos, or tan ] *b*(x - *c*) + *k* a = amplitude b = a number that determines your period c = phase shift ( horizontal shift ) k = vertical shift *midline* : y = k
If the terminal side of an angle makes more than one complete rotation, then . . .
you add 360° to the current degree
When finding out the exact value of a cosine or sine expresssion, then . . .
you cancel out full rotations ex: *cos 480°* = cos(*360* + 120) = cos(*120*) = -½
What is 360° and 180° in radians?
*2π* radians = 360° *π* radians = 180°
When evaluating trigonometric functions in certain quadrants, how do you memorize what functions are positive and what are negative?
*A*ll ( Q.1, all functions are positive ) *S*tudents ( Q.2, sine and its reciprocal are positive ) *T*ake ( Q.3, tangent and its reciprocal are positive ) *C*alculus ( Q.4, cosine and its reciprocal are positive )
How do you translate a trigonometric function utilizing a table?
*Rule* : Besides the first column, always use the values of the previous column to solve for the current column! -------------------------------------- 1. Construct a table of columns and rows, your first column will contain the parent theta values, of the trigonometric function. *Sine & Cosine* : 0°, 90°, 180°, 270°, 360° *Tangent* : -90°,-45°, 0°, 45°, 90° -------------------------------------- 2. Use the column to the right of the first to state the coordinate values of the trigonometric function *Sine* : 0, 1, 0, -1, 0 ( starts at the origin ) *Cosine* : 1, 0, -1, 0, 1 ( starts at highest point ) *Tangent* : und, -1, 0, 1, und ( passes through the origin ) -------------------------------------- 3. Use the other column to the right for the amplitude, which is the *a* term. Utilize the *a* term with the previous column to find the amplitude of each value. -------------------------------------- 4. Use another column to the right to factor in the vertical shift, which can either be added or subtracted from the previous column. -------------------------------------- 5. Now start going to the left of your first column. Start factoring in your *x* term, or the term that is behind the trigonometric function (cos, tan, sin, ect . . ) Simply, take everything that is behind the trigonometric function (cos, tan, sin, ect . . ) and set it equal to *x*, solving for theta. Every step you do to reach theta, make separate columns and do the same to the values of the previous column. -------------------------------------- 6. Utilize the last column to the left and the last column to the right to begin graphing
What are the trigonometric functions?
*Sin*e θ = opposite/hypotenuse *Cos*ine θ = adjacent/hypotenuse *Tan*gent θ = opposite/adjacent *Note* : The normal trigonometric functions can be remembered by the acronym *SOH CAH TOA* -------------------------------------- *c*o*s*e*c*ant θ = hypotenuse/opposite *sec*ant θ = hypotenuse/adjacent *cot*angent θ = adjacent/opposite
Where do the trigonometric functions start in the parent function?
*Sine* starts at the origin. *Cosine* starts at the highest point on the y - axis. *Tangent* curve passes through the origin.
What are the ambiguous cases of an acute triangle?
*a < h* : no solution *a = h* : one solution *h < a < b* : two solutions *a ≥ b* : one solution *Note* : h = b( sinA )
What are the ambiguous cases of an obtuse triangle?
*a ≤ b* : no solution *a > b* : one solution
You use inverse trigonometric ratios to find . . .
*angles* --------------------------- If sin A = x then sin⁻¹ x = A If sin 27° = 0.4540 then sin⁻¹( 0.4540 ) = 27°
What is the formula for finding circumference?
*c = πd* or *c = 2πr* *Note* : circumference / diameter = π
The Angle of Elevation is ________________ to the Angle of Depression
*equal*
What are the quadrant angles in both degrees and radians?
*θ = 0°* or *0 radians* *θ = 90°* or *π/2* *θ = 180°* or *π radians* *θ = 270°* or *3π/2*
Define *standard position*, *terminal side*, and *initial side*
- *Standard position* is the angle on the coordinate plane as its vertex on the origin and one ray is on the positive x-axis. - *Initial Side* : the ray that remains on the x-axis - *Terminal Side* : the ray of the angle that rotates between terminals
When do you use the Law of Sines?
- Two angles and any sides - Two sides and an angle opposite one of them *Note* : When you have a matching pair on a non-right triangle, which means you will know that the measure of one of the angles as well as its opposite side.
When do you use Law of Cosines?
- Two sides and their included angle - Three sides *Note* : When you have a non-right triangle and you do not have a matching pair.
If you are asked to find a positive and negative *coterminal* angle for a given angle, you . . .
- add 360° to the given measurement to get the positive angle - subtract 360° to the given measurement to get the negative angle - *However*, if you subtract 360°, but do not get a negative angle, you must keep subtracting 360° until the angle is negative ( ex : 630° [ + 360° = 990° ] [ *-360° = 270° - 360° = -90°* ] ) ex: 130° positive coterminal angle : 130° + 360° = 490° negative coterminal angle : 130° - 360° = -230°
What are the coordinate points of quadrant angles?
0° = ( 1,0 ) 90° = ( 0, 1 ) 180° = ( -1, 0 ) 270° = ( 0, -1 )
What are the useful memory conversions of degrees and radians?
30° = π/6 45° = π/4 60° = π/3 90° = π/2
How do you find the measures of the second triangle, if there are two solutions?
Find the supplement of angle 'B' and keep angle 'A' the same, then add angle 'A' and the supplement of angle 'B', and subtract that sum from 180 to find the measurement of new angle 'C'. Then find new side 'c'.
What is the Law of Cosines?
If triangle ABC has lengths a, b, and c, representing the lengths of the sides opposite the angles with measures A, B, and C, then *a*² = *b*² + *c*² - 2*bc* cos*A* *b*² = *a*² + *c*² - 2*ac* cos*B*, and *c*² = *a*² + *b*² - 2*ab* cos*C* (side looking-for)² = (side1)² + (side2)² - 2(side1)(side2) cos(angle opposite side looking-for)
What is the Law of Sines?
If triangle ABC has lengths a, b, and c, representing the lengths of the sides opposite the angles with measures A, B, and C, then *sinA/a = sinB/b = sinC/c*
What is the *30-60-90 Triangle Theorem*?
In a 30-60-90 triangle, the length of the hypotenuse is two times the length of the short leg and the long leg is √3 times the short leg *30-60-90* : *1* goes opposite 30°, *2* goes opposite 90°, and *√3* goes opposite 60°
What is the *45-45-90 Triangle Theorem*?
In a 45-45-90 triangle, the legs ( ℓ ) are congruent and the length of the hypotenuse is the length of the legs times √2 *45-45-90* : *1*s go on legs, and *√2* goes on hypotenuse
When is a hypotenuse present?
In a right triangle
Define reference angle
The angle inside of the triangle, closest to the origin.
What is the trigonometric area of a triangle?
The area of a triangle is one half the product of the lengths of two sides and the sine of their included angle.
When do you use the Pythagorean Theorem?
When you are finding the side of a right triangle
How do you graph reciprocals trigonometric functions?
They are graphed at the period or the highest point of every cycle; however, for cotangent, you just switch the direction of the curve. *Note* : Similar to tangent, reciprocal graphs do not have an amplitude, they just have the highest point
If the measure of an angle is negative, the terminal side is rotated _______________________________.
clockwise
If the measure of the an angle is positive, the terminal side is rotated _______________________________.
counterclockwise
What is the formula for finding the diameter of a circle?
d = 2r
