Trigonometry Packet (Last Unit)

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

Trigonometry Packet

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

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12-1 Study Guide and Intervention

Trigonometric Functions in Right Triangles

Trigonometric Functions for Acute Angles Trigonometry is the study of relationships among the angles and sides
of a right triangle. A trigonometric function has a rule given by a trigonometric ratio, which is a ratio that compares the
side lengths of a right triangle.

Trigonometric Functions If θ is the measure of an acute angle of a right triangle, opp is the measure of the leg
in Right Triangles opposite θ, adj is the measure of the leg adjacent to θ, and hyp is the measure of the

hypotenuse, then the following are true.

sin θ = opp cos θ = adj tan θ = opp

hyp hyp adj

csc θ = hyp sec θ = hyp cot θ = adj

opp adj opp

Example: In a right triangle, ∠ B is acute and cos B = . Find the value of tan B.
Step 1 Draw a right triangle and label one acute angle B. Label the adjacent side 3 and the hypotenuse 7.

Step 2 Use the Pythagorean Theorem to find b.

2 + 2 = 2 Pythagorean Theorem

32 + 2 = 72 a = 3 and c = 7

9 + 2 = 49 Simplify.

2 = 40 Subtract 9 from each side.

b = √40 or 2√10 Take the positive square root of each side.

Step 3 Find tan B.

tan B = opp Tangent function
adj

tan B = 2√10 Replace opp with 2√10 and adj with 3.
3

Exercises

Find the values of the six trigonometric functions for angle θ.

1. 2. 3.

In a right triangle, ∠ A and ∠ B are acute.

4. If tan A = 7 , what is cos A? 5. If cos A = 1 , what is tan A? 6. If sin B = 3 , what is tan B?
12 2 8

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12-1 Study Guide and Intervention (continued)

Trigonometric Functions in Right Triangles

Use Trigonometric Functions You can use trigonometric functions to find missing side lengths and missing angle
measures of right triangles. You can find the measure of the missing angle by using the inverse of sine, cosine, or tangent.

Example: Find the measure of ∠ C. Round to the nearest tenth if necessary.

You know the measure of the side opposite ∠ C and the measure of the hypotenuse. Use the sine function.

sin C = opp Sine function
hyp

sin C = 8 Replace opp with 8 and hyp with 10.
10 Inverse sine
Use a calculator.
sin−1 8 = m∠ C
10

53.1° ≈ m∠ C

Exercises

Use a trigonometric function to find each value of x. Round to the nearest tenth if necessary.

1. 2. 3.

4. 5. 6.

Find x. Round to the nearest tenth if necessary. 9.
7. 8.

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12-2 Study Guide and Intervention

Angles and Angle Measure

Angles in Standard Position An angle is determined by two rays. The degree measure of an angle in standard position
is described by the amount and direction of rotation from the initial side, which lies along the positive x-axis, to the
terminal side. A counterclockwise rotation is associated with positive angle measure and a clockwise rotation is
associated with negative angle measure. Two or more angles in standard position with the same terminal side are called
coterminal angles.

Example 1: Draw an angle with measure 290° in standard position.
The negative y-axis represents a positive rotation of 270°. To generate an angle of
290°, rotate the terminal side 20° more in the counterclockwise direction

Example 2:Find an angle with a positive measure and an angle with a negative measure that are coterminal
with each angle.

a. 250° Add 360°.

A positive angle is 250° + 360° or 610°. Subtract 360°.
A negative angle is 250° – 360° or –110°.

b. –140°

A positive angle is –140° + 360° or 220°. Add 360°.
A negative angle is –140° – 360° or –500°. Subtract 360°.

Exercises

Draw an angle with the given measure in standard position.

1. 160° 2. 280° 3. 400°

Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle.

4. 65° 5. –75° 6. 230° 7. 420°

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12-2 Study Guide and Intervention (continued)

Angles and Angle Measure

Convert Between Degrees and Radians Angles can be measured in degrees and radians, which are units based on
arc length. One radian is the measure of an angle θ in standard position with a terminal side that intercepts an arc with the
same length as the radius of the circle. Degree measure and radian measure are related by the equations 2π radians = 360°

and π radians = 180°.

Radian and To rewrite the radian measure of an angle in degrees, multiply the number of radians by 180° .
Degree
Measure π radians

To rewrite the degree measure of an angle in radians, multiply the number of degrees by π radians.

180°

Arc For a circle with radius r and central angle θ (in radians), the arc length s equals the product of r and θ.
Length s = rθ

Example 1: Rewrite each degree measure in radians Example 2: A circle has a radius of 5 cm and central
and the radian measure in degrees. angle of 135°, what is the length of the circle’s arc?

a. 45° Find the central angle in radians.

45° = 45° (π radians) = radians 135° = 135°(π radians) = 3 radians
4 4
180° 180°

Use the radius and central angle to find the arc length.

b. radians s = rθ Write the formula for arc length.


5 radians = 5 (180° ) = 300° = 5 ⋅ 3 Replace r with 5 and θ with 3
3 3 4 4

≈ 11.78 Use a calculator to simplify

Exercises
Rewrite each degree measure in radians and each radian measure in degrees.

1. 140° 2. –260° 3. – 3
5

4. –75° 5. 7 6. 380°
6

Find the length of each arc. Round to the nearest tenth. 9.
7. 8.

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12-8 Study Guide and Intervention

Translations of Trigonometric Graphs

Horizontal Translations When a constant is subtracted from the angle measure in a trigonometric function,
a phase shift of the graph results.

Phase Shift The phase shift of the graphs of the functions y = a sin b(θ – h), y = a cos b(θ – h),
and y = a tan b(θ – h) is h, where b > 0.

If h > 0, the shift is h units to the right.

If h < 0, the shift is |h| units to the left.

Example: Find State the amplitude, period, and phase shift

for y = cos 3 ( – ). Then graph the function.


Amplitude: |a| = | 1 | or 1
2 2

Period: 2 = 2 or 2
| | |3| 3

Phase Shift: h =
2

The phase shift is to the right since 2 > 0.

Exercises
State the amplitude, period, and phase shift for each function. Then graph the function.

1. y = 2 sin (θ + 60°) 2. y = tan ( – 2 )

3. y = 3 cos (θ – 45°) 4. y = 1 sin 3 ( – 3 )
2

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12-8 Study Guide and Intervention (continued)

Translations of Trigonometric Graphs

Vertical Translations When a constant is added to a trigonometric function, the graph is shifted vertically.

Vertical Shift The vertical shift of the graphs of the functions y = a sin b(θ – h) + k, y = a cos b(θ – h) + k,
and y = a tan b(θ – h) + k is k.

If k > 0, the shift is k units up.

If k < 0, the shift is |k| units down.

The midline of a vertical shift is y = k.

Graphing Step 1 Determine the vertical shift, and graph the midline.
Trigonometric Step 2 Determine the amplitude, if it exists. Use dashed lines to indicate the maximum and
Functions
minimum values of the function.
Step 3 Determine the period of the function and graph the appropriate function.

Step 4 Determine the phase shift and translate the graph accordingly.

Example: State the amplitude, period, vertical shift, and equation of the
midline for y = cos 2θ – 3. Then graph the function.

Amplitude: |a| = |1| or 1

Period: 2 = 2 or π
| | |2|

Vertical Shift: k = –3, so the vertical shift is 3 units down.

The equation of the midline is y = –3.

Since the amplitude of the function is 1, draw dashed lines

parallel to the midline that are 1 unit above and below the midline.
Then draw the cosine curve, adjusted to have a period of π.

Exercises
State the amplitude, period, vertical shift, and equation of the midline for each function. Then graph the function.

1. y = 1 cos θ + 2 2. y = 3 sin θ – 2
2

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13-1 Study Guide and Intervention

Trigonometric Identities

Find Trigonometric Values A trigonometric identity is an equation involving trigonometric functions that is true for
all values for which every expression in the equation is defined.

Basic Quotient Identities tan = sin cot = cos cot = 1
Trigonometric Reciprocal Identities
Identities Pythagorean Identities cos sin tan

cos = 1 sec = 1

sin cos

cos2 + sin2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2

Example: Find the exact value of cot θ if csc θ = − and 180° < θ < 270°.


cot2 θ + 1 = csc2 θ Trigonometric identity

cot2 θ + 1 = (− 151)2 Substitute − 11 for csc θ.
5

cot2 θ + 1 = 121 Square − 11.
25
5

cot2 θ = 96 Subtract 1 from each side.
25

cot θ = ± 4√6 Take the square root of each side.
5

Since θ is in the third quadrant, cot θ is positive. Thus, cot θ = 4√56.

Exercises
Find the exact value of each expression if 0° < θ < 90°.

1. If cot θ = 4, find tan θ. 2. If cos θ = √23, find csc θ.

3. If sin θ = 53, find cos θ. 4. If sin θ = 13, find sec θ.

5. If tan θ = 34, find cos θ. 6. If sin θ = 73, find tan θ.

Find the exact value of each expression if 90° < θ < 180°. 8. If csc θ = 152, find cot θ.
7. If cos θ = − 78, find sec θ.

Find the exact value of each expression if 270° < θ < 360°. 10. If csc θ = − 49, find sin θ.
9. If cos θ = 67, find sin θ.

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13-1 Study Guide and Intervention (continued)

Trigonometric Identities

Simplify Expressions The simplified form of a trigonometric expression is written as a numerical value or in terms
of a single trigonometric function, if possible. Any of the trigonometric identities can be used to simplify expressions
containing trigonometric functions.

Example 1: Simplify (1 – θ) sec θ cot θ + tan θ sec θ θ.

(1 – cos2 θ) sec θ cot θ + tan θ sec θ cos2 θ = sin2 θ · 1 θ · cos + sin · 1 · cos2 θ
cos sin cos cos
= sin θ + sin θ

= 2 sin θ

Example 2: Simplify · – .
− +

sec · cot – csc = 1 ⋅ cos 1
1 − sin + sin cos sin
– sin
1 1 + sin
1 + sin

=1 (1 + sin ) − 1 (1 − sin )
sin sin

(1 − sin )(1 + sin )

= 1 + 1 − 1 + 1
sin sin

1 − sin2

= 2 or 2 sec2 θ
cos2

Exercises
Simplify each expression.

1. tan · csc 2. sin · cot
sec sec2 − tan2


3. sin2 − cot · tan 4. cos
cot · sin − tan
sec

5. tan · cos + cot θ · sin θ · tan θ · csc θ 6. csc2 − cot2
sin tan · cos

7. 3 tan θ · cot θ + 4 sin θ · csc θ + 2 cos θ · sec θ 8. 1 − cos2
tan · sin

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