5-1 Study Guide and Intervention - MRS. FRUGE · 5-1 Study Guide and Intervention Trigonometric Identities Basic Trigonometric Identities An equation is an identity if the left side

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

Chapter 5 5 Glencoe Precalculus

5-1 Study Guide and Intervention Trigonometric Identities

Basic Trigonometric Identities An equation is an identity if the left side is equal to the right side for all values of the

variable for which both sides are defined. Trigonometric identities are identities that involve trigonometric functions.

Reciprocal Identities Pythagorean Identities

sin θ = 1

csc θ

csc θ = 1

sin 𝜃

cos θ = 1

sec 𝜃

sec θ = 1

cos 𝜃

tan θ = 1

cot 𝜃

cot θ = 1

tan 𝜃

sin2 θ + cos2 θ = 1

tan2 θ + 1 = sec2 θ

cot2 θ + 1 = csc2 θ

Example: If sin θ = 𝟑

𝟓 and 0° < θ < 90°, find tan θ.

Use two identities to relate sin θ and tan θ.

sin2 θ + cos2 θ = 1 Pythagorean Identity

(3

5)

2

+ cos2 θ = 1 sin θ = 3

5

cos2 θ = 16

25 Simplify.

cos θ = ± √16

25 or ±

4

5 Take the square root of each side.

Since 0° < θ < 90°, cos θ is positive.

Thus, cos θ = 4

5.

Now find tan θ.

tan θ = sin θ

cos θ Quotient identity

tan θ =

3

54

5

sin θ = 3

5, cos θ =

4

5

tan θ = 3

4 Simplify.

Exercises

Find the value of each expression using the given information.

1. If cot θ = 12

5, find tan θ. 2. If sin θ = –

1

4, find csc θ.

3. If tan α = 2

3, find cot α. 4. If sec β = –2, find csc (𝛽 −

𝜋

2).

5. If cot α = – 4

3 and sin α < 0, find cos α and csc α.

6. If sec α = –4 and csc α > 0, find cos α and tan α.

𝟓

𝟏𝟐 –4

𝟑

𝟐 2

cos α = 𝟒

𝟓, and csc α = –

𝟓

𝟑

cos α = – 𝟏

𝟒, and tan α = –√𝟏𝟓

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-1 Study Guide and Intervention (continued)

Trigonometric Identities

Simplify and Rewrite Trigonometric Expressions You can apply trigonometric identities and algebraic techniques

such as substitution, factoring, and simplifying fractions to simplify and rewrite trigonometric expressions.

Example: Simplify each expression.

a. sec x – cos x

sec x − cos x = 1

cos 𝑥 − cos x Reciprocal Identity

= 1 − cos2 𝑥

cos 𝑥 Add.

= sin2 𝑥

cos 𝑥 Pythagorean Identity

= sin x (sin 𝑥

cos 𝑥) Factor.

= sin x tan x Quotient Identity

b. csc x 𝐜𝐨𝐭𝟐 x + 𝟏

𝐬𝐢𝐧 𝒙

csc x cot2 x + 1

sin 𝑥 = csc x cot2 x + csc x Reciprocal Identity

= csc x (csc2 x – 1) + csc x Pythagorean Identity

= csc3 x – csc x + csc x Distributive Property

= csc3 x Simplify.

Exercises

Simplify each expression.

1. cos x (tan x + cot x) 2. sin x + cos x cot x

3. csc2 𝑥

1 + tan2 𝑥 4. (sec x – tan x)(csc x + 1)

5. (cot2 x + 1)(sec2 x – 1) 6. 1 + tan2 𝑥

1 + sec 𝑥

7. csc x sin x + cot2 x 8. cos x (1 + tan2 x )

9. cos (

π

2 – 𝑥)

csc 𝑥 10.

cos (π

2 – 𝑥)

csc 𝑥 + cos2 x

csc x csc x

𝐜𝐨𝐭𝟐 x cot x

𝐬𝐞𝐜𝟐 x sec x

𝐜𝐬𝐜𝟐 x sec x

𝐬𝐢𝐧𝟐 x 1

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-1 Practice Trigonometric Identities

Find the value of each expression using the given information.

1. If cos θ = 1

4 and 0˚ < θ < 90˚, find tan θ. 2. If sin θ =

2

3 and 0˚ < θ < 90˚, find cos θ.

3. If tan θ = 7

2 and 0˚ < θ < 90˚, find sin θ. 4. If tan θ = 2 and 0˚ < θ < 90˚, find cot θ.

5. If sin α = – 4

5 and cos α > 0, find tan α and sec α.

6. If cot x = – 3

2 and sec x < 0, find sin x and cos x.

7. If cos θ = 0.54, find sin (𝜃 − 𝜋

2).

8. If cot x = –0.18, find tan (𝑥 − 𝜋

2).


9. cos x + sin x tan x 10. cot 𝐴

tan 𝐴

11. sin2 θ cos2 θ – cos2 θ 12. csc2 𝑥 − cot2 𝑥

sin (−𝑥) cot 𝑥

13. KITE FLYING Brett and Tara are flying a kite. When the string is tied to the ground, the height of the kite can be

determined by the formula 𝐿

𝐻 = csc θ, where L is the length of the string and θ is the angle between the string and the

level ground. What formula could Brett and Tara use to find the height of the kite if they know the value of sin θ?

√𝟏𝟓 √𝟓

𝟑

𝟕√𝟓𝟑

𝟓𝟑

𝟏

𝟐

tan α = – 𝟒

𝟑, sec α =

𝟓

𝟑

sin x = 𝟐√𝟏𝟑

𝟏𝟑, cos x = –

𝟑√𝟏𝟑

𝟏𝟑

–0.54

0.18

sec x –𝐜𝐨𝐭𝟐 A

–𝐜𝐨𝐬𝟒 θ –sec x

H = L sin θ

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-2 Study Guide and Intervention Verifying Trigonometric Identities

Verify Trigonometric Identities To verify an identity means to prove that both sides of the equation are equal for all

values of the variable for which both sides are defined.

Example: Verify that 𝐬𝐞𝐜𝟐 𝒙 – 𝟏

𝐬𝐞𝐜𝟐 𝒙 = 𝐬𝐢𝐧𝟐 x.

The left-hand side of this identity is more complicated, so start with that expression first.

sec2 𝑥 – 1

sec2 𝑥 =

(tan2 𝑥 + 1) − 1

sec2 𝑥 Pythagorean Identity

= tan2 𝑥

sec2 𝑥 Simplify.

= (

sin2 𝑥

cos2 𝑥)

1

cos2 𝑥

Quotient Identity and Reciprocal Identity

= sin2 𝑥

cos2 𝑥 · cos2 x Simplify.

= sin2 x✓ Multiply.

Notice that the verification ends with the expression on the other side of the identity.

Exercises

Verify each identity.

1. sec θ – cos θ = sin θ tan θ

2. sec θ = sin θ(tan θ + cot θ)

3. tan θ csc θ cos θ = 1

4. csc2 𝜃 − cot2 𝜃

1 − sin2 𝜃 = sec2 θ

sec θ − cos θ = 𝟏

𝒄𝒐𝒔 𝜽 – cos θ =

𝟏 – 𝐜𝐨𝐬𝟐𝜽

𝐜𝐨𝐬 𝜽 =

𝐬𝐢𝐧𝟐𝜽

𝐜𝐨𝐬 𝜽 = sin θ (

𝐬𝐢𝐧 𝜽

𝐜𝐨𝐬 𝜽)

= sin θ tan θ

sin θ (tan θ + cot θ) = sin θ (𝐬𝐢𝐧 𝜽

𝐜𝐨𝐬 𝜽+

𝐜𝐨𝐬 𝜽

𝐬𝐢𝐧 𝜽) = sin θ(

𝐬𝐢𝐧𝟐 𝜽 + 𝐜𝐨𝐬𝟐 𝜽

𝒄𝒐𝒔 𝜽 𝒔𝒊𝒏 𝜽)

= sin θ (𝟏

𝐜𝐨𝐬 𝜽 𝐬𝐢𝐧 𝜽)=

𝟏

𝐜𝐨𝐬 𝜽 = sec θ

tan θ csc θ cos θ = (𝐬𝐢𝐧 𝜽

𝐜𝐨𝐬 𝜽) (

𝟏

𝐬𝐢𝐧 𝜽) cos θ = 1

𝐜𝐬𝐜𝟐𝜽 − 𝐜𝐨𝐭𝟐𝜽

𝟏 – 𝐬𝐢𝐧𝟐 𝜽 =

(𝐜𝐨𝐭𝟐 𝜽 + 𝟏) – 𝐜𝐨𝐭𝟐 𝜽

𝐜𝐨𝐬𝟐 𝜽 =

𝟏

𝐜𝐨𝐬𝟐 𝜽 = 𝐬𝐞𝐜𝟐 𝜽

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Verifying Trigonometric Identities

Identifying Identities and Nonidentities You can use a graphing calculator to test whether an equation might be an

identity by graphing the functions related to each side of the equation. If the graphs of the related functions do not coincide for all values of x for which both functions are defined, the equation is not an identity. If the graphs appear to

coincide, you can verify that the equation is an identity by using trigonometric properties and algebraic techniques.

Example: Use a graphing calculator to test whether csc θ – sin θ = cot θ cos θ is an identity. If it appears to be an

identity, verify it. If not, find an x-value for which both sides are defined but not equal.

The equation appears to be an identity because the graphs of the related functions

coincide. Verify this algebraically.

csc θ – sin θ = 1

sin 𝜃 – sin θ Rewrite in terms of sine using a Reciprocal Identity.

= 1 − sin2 𝜃

sin 𝜃 Rewrite using a common denominator.

= cos2 𝜃

sin 𝜃 Pythagorean Identity

= cos 𝜃

sin 𝜃 · cos θ Factor cos2 θ.

= cot θ cos θ ✓ Rewrite in terms of cot θ using a Quotient Identity.

Exercises

Test whether each equation is an identity by graphing. If it appears to be an identity, verify it.

If not, find an x-value for which both sides are defined but not equal.

1. sin x + cos x cot x = csc x 2. 2 – cos2 x = sin2 x

sin x + cos x cot x When x = 𝝅

𝟒, 𝐘𝟏 = 1.5

= sin x + cos x (𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙) and 𝐘𝟐 = 0.5;

= sin x + 𝐜𝐨𝐬𝟐 𝒙

𝐬𝐢𝐧 𝒙 =

𝐬𝐢𝐧𝟐 𝒙 + 𝒄𝒐𝒔𝟐 𝒙

𝐬𝐢𝐧 𝒙 therefore, the equation is not an identity.

= 𝟏

𝐬𝐢𝐧 𝒙 = csc x

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-2 Practice Verifying Trigonometric Identities

Verify each identity.

1. csc 𝑥

cot 𝑥 + tan 𝑥 = cos x 2.

1

sin 𝑦 − 1 −

1

sin 𝑦 + 1 = −2 sec2 y

3. sin3 x – cos3 x = (1 + sin x cos x)(sin x – cos x) 4. tan θ + cos 𝜃

1 + sin 𝜃 = sec θ

5. (sec θ − tan θ)2 = 1 − sin 𝜃

1 + sin 𝜃 6.

sin 𝜃

1 + cos 𝜃 +

1 + cos 𝜃

sin 𝜃 = 2 csc θ

Test whether each equation is an identity by graphing. If it appears to be an identity, verify it.

If not, find an x-value for which both sides are defined but not equal.

7. cos 𝑥

1− sin 𝑥 =

1 + sin 𝑥

cos 𝑥 8. sin x(sec x + cot x) = cos x

9. PHYSICS The work done in moving an object is given by the formula W = Fd cos θ, where d is the displacement, F is

the force exerted, and θ is the angle between the displacement and the force. Verify that W = 𝐹𝑑 cot 𝜃

csc 𝜃 is an equivalent

formula.

𝐜𝐬𝐜 𝒙

𝐜𝐨𝐭 𝒙 +𝐭𝐚𝐧 𝒙 =

𝟏

𝐬𝐢𝐧 𝒙𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙 +

𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙

• 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙

𝟏

𝐬𝐢𝐧 𝒚−𝟏 –

𝟏

𝐬𝐢𝐧 𝒚+𝟏 =

𝐬𝐢𝐧 𝒚 + 𝟏 – 𝐬𝐢𝐧 𝒚 + 𝟏

𝐬𝐢𝐧𝟐 𝒚 – 𝟏

= 𝐜𝐨𝐬 𝒙

𝐜𝐨𝐬𝟐 𝒙 + 𝐬𝐢𝐧𝟐 𝒙 =

𝟐

–𝐜𝐨𝐬𝟐 𝒚

= 𝐜𝐨𝐬 𝒙

𝟏 = 𝐜𝐨𝐬 𝒙 ✓ = –2 𝐬𝐞𝐜𝟐 y ✓

𝐬𝐢𝐧𝟑x – 𝐜𝐨𝐬𝟑 x = (sin x – cos x) tan θ + 𝐜𝐨𝐬 𝜽

𝟏+𝐬𝐢𝐧 𝜽 =

𝐬𝐢𝐧 𝜽

𝐜𝐨𝐬 𝜽 +

𝒄𝒐𝒔 𝜽

𝟏 + 𝐬𝐢𝐧 𝛉

(𝐬𝐢𝐧𝟐 x + sin x cos x + 𝐜𝐨𝐬𝟐 x) = 𝐬𝐢𝐧 𝜽+ 𝐬𝐢𝐧𝟐 𝜽+ 𝐜𝐨𝐬𝟐 𝜽

(𝐜𝐨𝐬 𝜽)(𝟏+𝐬𝐢𝐧 𝜽)

= (sin x – cos x)(1 + sin x cos x) = 𝟏+𝐬𝐢𝐧 𝜽

(𝐜𝐨𝐬 𝜽 )(𝟏+𝐬𝐢𝐧 𝜽) = sec θ ✓

= (1 + sin x cos x)(sin x – cos x) ✓

(𝐬𝐞𝐜 𝜽 – 𝐭𝐚𝐧 𝜽)𝟐 = 𝐬𝐞𝐜𝟐 θ – 2 sec θ tan θ + 𝐭𝐚𝐧𝟐 θ

= 𝟏

𝐜𝐨𝐬𝟐 𝜽 – 2(

𝟏

𝐜𝐨𝐬 𝜽) (

𝐬𝐢𝐧 𝜽

𝐜𝐨𝐬 𝜽) +

𝐬𝐢𝐧𝟐𝜽

𝐜𝐨𝐬𝟐𝜽

𝐬𝐢𝐧 𝜽

𝟏+𝐜𝐨𝐬 𝜽 +

𝟏+𝐜𝐨𝐬 𝜽

𝐬𝐢𝐧 𝛉 =

𝐬𝐢𝐧𝟐 𝜽+𝟏+𝟐 𝐜𝐨𝐬 𝜽+𝐜𝐨𝐬𝟐 𝜽

𝐬𝐢𝐧 𝜽 (𝟏 + 𝐜𝐨𝐬 𝜽)

= 𝟏 – 𝟐 𝐬𝐢𝐧 𝜽 + 𝐬𝐢𝐧𝟐 𝜽

𝐜𝐨𝐬𝟐 = 𝟐 + 𝟐 𝐜𝐨𝐬 𝜽


= (𝟏 – 𝐬𝐢𝐧 𝜽)𝟐

𝟏–𝐬𝐢𝐧𝟐 𝜽 =

(𝟏 – 𝐬𝐢𝐧 𝜽)𝟐

(𝟏 + 𝐬𝐢𝐧 𝜽)(𝟏 – 𝐬𝐢𝐧 𝜽) =

𝟏 – 𝐬𝐢𝐧 𝜽

𝟏 + 𝐬𝐢𝐧 𝜽 ✓ =

𝟐 (𝟏 + 𝐜𝐨𝐬 𝜽)


= 𝟐

𝐬𝐢𝐧 𝜽 = 2 csc θ ✓

𝐜𝐨𝐬 𝒙

𝟏 – 𝐬𝐢𝐧 𝒙 =

𝐜𝐨𝐬 𝒙

𝟏 – 𝐬𝐢𝐧 𝒙 •

𝟏 + 𝐬𝐢𝐧 𝒙

𝟏 + 𝐬𝐢𝐧 𝒙 When x =

𝝅

𝟒, 𝐘𝟏= 1.7 and 𝐘𝟐= 0.7;

= 𝐜𝐨𝐬 𝒙 (𝟏 + 𝐬𝐢𝐧 𝒙)

𝟏 – 𝐬𝐢𝐧𝟐 𝒙 therefore, the equation is not an identity.

= 𝐜𝐨𝐬 𝒙 (𝟏 + 𝐬𝐢𝐧 𝒙)

𝐜𝐨𝐬𝟐 𝒙

= 𝟏 + 𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙 ✓

W = 𝑭𝒅 𝐜𝐨𝐭 𝜽

𝐜𝐬𝐜 𝜽 =

𝑭𝒅(𝐜𝐨𝐬 𝜽

𝐬𝐢𝐧 𝜽)

𝟏

𝐬𝐢𝐧 𝜽

= 𝑭𝒅 𝐜𝐨𝐬 𝜽

𝐬𝐢𝐧 𝜽 • sin θ = Fd cos θ

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-3 Study Guide and Intervention Solving Trigonometric Equations

Use Algebraic Techniques to Solve To solve a trigonometric equation, you may need to apply algebraic methods.

These methods include isolating the trigonometric expression, taking the square root of each side, factoring and applying the Zero-Product Property, applying the quadratic formula, or rewriting using a single trigonometric function. In this

lesson, we will consider conditional trigonometric equations, or equations that may be true for certain values of the

variable but false for others.

Example 1: Find all solutions of tan x cos x – cos x = 0 on the interval [0, 2π).

tan x cos x – cos x = 0 Original equation

cos x (tan x – 1) = 0 Factor.

cos x = 0 or tan x – 1 = 0 Set each factor equal to 0.

x = 𝜋

2 or

3𝜋

2 tan x = 1

x = 𝜋

4 or

5𝜋

4

When x = 𝜋

2 or

3𝜋

2, tan x is undefined, so the solutions of the original equation are

𝜋

4 or

5𝜋

4. When you solve for all values of

x, the solution should be represented as x + 2nπ for sin x and cos x and x + nπ for tan x, where n is any integer. The

solutions are 𝜋

4 + nπ or

5𝜋

4 + nπ.

Example 2: Find all solutions of sin x + √𝟑 = –sin x.

sin x + √3 = –sin x Original equation

2 sin x + √3 = 0 Add sin x to each side.

2 sin x = – √3 Subtract √3 from each side.

sin x = – √3

2 Divide each side by 2.

x = 4𝜋

3 or

5𝜋

3 Solve for x.

The solutions are 4𝜋

3 + 2nπ or

5𝜋

3 + 2nπ.

Exercises

Solve each equation for all values of x.

1. cos x = –1 2. sin3 x – 4 sin x = 0

3. sin x cos x –3 cos x = 0 4. 2 sin3 x = sin x

Find all solutions of each equation on the interval [0, 2π).

5. 2 cos x = 1 6. 5 + 2 sin x – 7 = 0

7. 4 sin2 x tan x = tan x 8. 2 cos x – √3 = 0

𝝅 + 2n𝝅 n𝝅

𝝅

𝟐 + n𝝅 n𝝅,

𝝅

𝟒 + n𝝅,

𝟑𝝅

𝟒 + n𝝅

𝝅

𝟑,

𝟓𝝅

𝟑

𝝅

𝟐

0, 𝝅

𝟔,

𝟓𝝅

𝟔, 𝝅,

𝟕𝝅

𝟔,

𝟏𝟏𝝅

𝟔

𝝅

𝟔,

𝟏𝟏𝝅

𝟔

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Solving Trigonometric Equations

Use Trigonometric Identities to Solve You can use trigonometric identities along with algebraic methods to solve

trigonometric equations. Be careful to check all solutions in the original equation to make sure they are valid solutions.

Example 1: Find all solutions of 2 𝐭𝐚𝐧𝟐 x – 𝐬𝐞𝐜𝟐 x + 3 = 1 – 2 tan x on the interval [0, 2π).

2 tan2 x – sec2 x + 3 = 1 – 2 tan x Original equation

2 tan2 x – (tan2 x + 1) + 3 = 1 – 2 tan x sec2 x = tan2 x + 1

tan2 x + 2 = 1 – 2 tan x Simplify.

tan2 x + 2 tan x + 1 = 0 Simplify.

(tan 𝑥 + 1)2 = 0 Factor.

tan x = –1 Take the square root of each side.

x = 3𝜋

4 or

7𝜋

4 Solve for x on [0, 2π).

Example 2: Find all solutions of 1 + cos x = sin x on the interval [0, 2π).

1 + cos x = sin x Original equation

(1 + cos 𝑥)2 = (sin x)2 Square each side.

1 + 2 cos x + cos2 x = sin2 x Multiply.

1 + 2 cos x + cos2 x = 1 – cos2 x Pythagorean Identity

2 cos2 x + 2 cos x = 0 Simplify.

2 cos x (cos x + 1) = 0 Factor.

cos x = 0 or cos x = –1 Zero Product Property

x = 𝜋

2, π,

3𝜋

2 Solve for x on [0, 2π).

Exercises


1. tan2 x = 1 2. 2 sin2 x – cos x = 1

3. sin x cos x –3 cos x = 0 4. cos2 x + sin x + 1 = 0


5. cos x = sin x 6. √3 cos x tan x – cos x = 0

7. tan2 x + sec x – 1 = 0 8. 1 + cos x = √3 sin x

𝝅

𝟒 +

𝒏𝝅

𝟐

𝝅

𝟑 + 2n𝝅,

𝟓𝝅

𝟑 + 2n𝝅, 𝝅 + 2n𝝅

𝝅

𝟐 + n𝝅

𝟑𝝅

𝟐 + 2n𝝅

𝝅

𝟒,

𝟓𝝅

𝟒

𝝅

𝟔,

𝟕𝝅

𝟔

0, 𝟐𝝅

𝟑,

𝟒𝝅

𝟑

𝝅

𝟑, 𝝅

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-3 Practice Solving Trigonometric Equations


1. cos x = 3 cos x – 2 2. 2 sin2 x – 1 = 0

3. √cos 𝑥 = 2 cos x – 1 4. 2 sin2 x – 5 sin x + 2 = 0


5. sec2 x + tan x = 1 6. 3 tan x – √3 = 0

7. 4 sin2 x – 4 sin x + 1 = 0 8. 4 cos2 x – 1 = 0

9. cos3 𝑥

sin 𝑥 = cot x 10. tan x sin2 x = 3 tan x

11. CIRCLES To find the diameter d of any circle, first inscribe a triangle in the circle. The diameter is then equal to the

ratio of any side of the triangle and the sine of its opposite angle.

a. Suppose the measure of one side of a triangle inscribed in a circle is 20 centimeters. If the measure of the angle in

the triangle opposite this side is 30°, what is the length of the diameter of the circle?

b. Suppose a circle with a diameter of 12.4 inches circumscribes a triangle with one side of the triangle measuring 4.6 inches. What is the measure of the angle in the triangle opposite this side?

12. AVIATION An airplane takes off from the ground and reaches a height of 500 feet after flying 2 miles. Given the

formula H = d tan θ, where H is the height of the plane and d is the distance (along the ground) the plane has flown,

find the angle of ascent θ at which the plane took off.

0 + 2n𝝅 𝝅

𝟒 +

𝒏𝝅

𝟐

0 + 2n𝝅 𝝅

𝟔 + 2n𝝅,

𝟓𝝅

𝟔 + 2n𝝅

0, 𝟑𝝅

𝟒, 𝝅,

𝟕𝝅

𝟒

𝝅

𝟔,

𝟕𝝅

𝟔

𝝅

𝟔,

𝟓𝝅

𝟔

𝝅

𝟑,

𝟐𝝅

𝟑,

𝟒𝝅

𝟑,

𝟓𝝅

𝟑

𝝅

𝟐,

𝟑𝝅

𝟐 0, 𝝅

40 cm

≈21.8°

about 2.7°

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-4 Study Guide and Intervention Sum and Difference Identities

Evaluate Trigonometric Functions You can use the sum and difference identities and the values of trigonometric

functions of common angles to find the exact values of less common angles.

Sum Identities

cos (α + β) = cos α cos β – sin α sin β

sin (α + β) = sin α cos β + cos α sin β

tan (α + β) = tan 𝛼 + tan 𝛽

1 − tan 𝛼 tan 𝛽

Difference Identities

cos (α – β) = cos α cos β + sin α sin β

sin (α – β) = sin α cos β – cos α sin β

tan (α – β) = tan 𝛼 − tan 𝛽

1+ tan 𝛼 tan 𝛽

Example: Find the exact value of cos 375°.

cos 375° = cos (330° + 45°) 330° and 45° are common angles with a sum of 375°.

= cos 330° cos 45° – sin 330° sin 45° Cosine Sum Identity

= √3

2 ·

√2

2 – (−

1

2) ·

√2

2 cos 330° =

√3

2, cos 45° =

√2

2, sin 330° = –

1

2, sin 45° =

√2

2

= √6

4 +

√2

4 Multiply.

= √6 + √2

4 Combine the fractions.

Exercises

Find the exact value of each trigonometric expression.

1. cos (–15°) 2. tan 15°

3. cos (− 7𝜋

12) 4. cos

11𝜋

12

5. sin 20° cos 10° + cos 20° sin 10° 6. tan

π

9 + tan

5𝜋

36

1 − tan π

9 tan

5𝜋

36


7. cos 70° cos 20° – sin 70° sin 20° 8. sin 𝜋

12 cos y – sin y cos

𝜋

12

Write each trigonometric expression as an algebraic expression.

9. cos (arcsin x + arccos x) 10. sin (arctan √3 – arcsin x)

11. Verify sin (π – x) = sin x. 12. Verify sin (270° + θ) = – cos θ.

√𝟔+ √𝟐

𝟒

√𝟑 –𝟏

𝟏+ √𝟑

√𝟐− √𝟔

𝟒

–√𝟔 – √𝟐

𝟒

𝟏

𝟐 1

cos 90° = 0 sin (𝝅

𝟏𝟐 – 𝒚)

0 √𝟑(𝟏−𝒙𝟐)− 𝒙

𝟐

sin (𝝅 – x) = sin 𝝅 cos x sin (270° + θ) = sin 270° cos θ – cos 𝝅 sin x = 0 • cos x + cos 270° sin θ = –1 cos θ

– (–1) • sin x = sin x ✓ + 0 • sin θ = – cos θ ✓

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Sum and Difference Identities

Solve Trigonometric Equations You can solve trigonometric equations using the sum and difference identities along

with algebraic methods and the same techniques you used before.

Example: Find the solutions of sin (𝝅

𝟐 + 𝒙) + cos (

𝝅

𝟐 + 𝒙) = 0 on the interval [0, 2π).

sin (𝜋

2 + 𝑥) + cos (

𝜋

2 + 𝑥) = 0 Original equation

sin 𝜋

2 cos x + cos

𝜋

2 sin x + cos

𝜋

2 cos x – sin

𝜋

2 sin x = 0 Cosine Sum Identity

1 (cos x) + 0 (sin x) + 0 (cos x) – 1 (sin x) = 0 Substitute.

cos x – sin x = 0 Simplify.

cos x = sin x Add.

On the interval [0, 2π), cos x = sin x when x = 𝜋

4 and x =

5𝜋

4.

Exercises

Find the solution of each equation on the interval [0, 2π).

1. cos (𝜋

4− 𝑥) – sin (

𝜋

4− 𝑥) = –1 2. sin (π + x) + sin (π + x) = 1

3. cos (3𝜋

2 + 𝑥) + sin (

3𝜋

2− 𝑥) = 0 4. tan (π – x) + tan (π – x) = –2

5. sin (𝑥 + 𝜋

3) + sin (𝑥 −

𝜋

3) = 1 6. cos (𝑥 +

𝜋

6) – 1 = cos (𝑥 −

𝜋

6)

𝟓𝝅

𝟒,

𝟕𝝅

𝟒

𝟕𝝅

𝟔,

𝟏𝟏𝝅

𝟔

𝝅

𝟒,

𝟓𝝅

𝟒

𝝅

𝟒,

𝟓𝝅

𝟒

𝝅

𝟐

𝟑𝝅

𝟐

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-4 Practice Sum and Difference Identities

Find the exact value of each trigonometric expression.

1. cos 5𝜋

12 2. sin (–165°) 3. tan 345°

4. csc 915° 5. tan (− 7𝜋

12) 6. sec

𝜋

12


7. cos 3𝜋

2 cos π – sin

3𝜋

2 sin π 8.

tan 30° − tan 𝑥

1 + tan 30° tan 𝑥

Write each trigonometric expression as an algebraic expression.

9. sin (arccos x + arcsin x) 10. cos (arccos 1

2 − arcsin 𝑥)

Verify each cofunction or reduction identity.

11. sin (360° + θ) = sin θ 12. cos (180° – θ) = –cos θ

Find the solutions to each expression on the interval [0, 2π).

13. cos (5π

4 + 𝑥) + sin (

5π

4− 𝑥) = 0 14. sin (

2π

3− 𝑥) + sin (

2π

3 + 𝑥) = 0

15. Sound waves can be modeled by the equations of the form 𝑦1 = 20 sin (3x + θ). A wave traveling in the opposite

direction can be modeled by 𝑦2 = 20 sin (3x – θ). Show that 𝑦1 + 𝑦2 = 40 sin 3x cos θ.

√𝟔– √𝟐

𝟒

√𝟐– √𝟔

𝟒 √𝟑 – 2

– √𝟔 – √𝟐 2 + √𝟑 √𝟔 – √𝟐

cos 𝟓𝝅

𝟐 tan (30° – x)

1 √𝟏– 𝒙𝟐 + 𝒙√𝟑

𝟐

sin (360° + θ) cos (180° – θ) = sin 360° cos θ + cos 360° sin θ = cos 180° cos θ + sin 180° sin θ = 0 • cos θ + 1 • sin θ = (–1) cos θ + 0 • sin θ = sin θ ✓ = –cos θ ✓ 𝝅

𝟒,

𝟓𝝅

𝟒

𝝅

𝟐,

𝟑𝝅

𝟐

𝒚𝟏+ 𝒚𝟐 = 20 sin (3x + θ) + 20 sin (3x – θ) = 20 (sin 3x cos θ + cos 3x sin θ) + 20 (sin 3x cos θ – cos 3x sin θ) = 40 sin 3x cos θ ✓

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-5 Study Guide and Intervention Multiple-Angle and Product-to-Sum Identities

Use Multiple-Angle Identities By letting α and β both equal θ in each of the angle sum identities you have learned

before, you can derive the following double-angle identities. The double-angle identities can then be used to derive the

power-reducing identities.

Double-Angle Identities

sin 2θ = 2 sin θ cos θ

tan 2θ = 2 tan 𝜃

1 − tan2 𝜃

cos 2θ = cos2 θ – sin2 θ

cos 2θ = 2 cos2 θ – 1

cos 2θ = 1 – 2 sin2 θ

Power-Reducing Identities

sin2 θ = 1 − cos 2𝜃

2 cos2 θ =

1 + cos 2𝜃

2 tan2 θ =

1 − cos 2𝜃

1 + cos 2𝜃

Example: If sin θ = 𝟏

𝟒 on the interval [0, 90°], find the exact value of sin 2θ.

To use the double-angle identity for sin 2θ, we must first find cos θ.

sin2 θ + cos2 θ = 1 Pythagorean Identity

(1

4)

2

+ cos2 θ = 1 Substitute 1

4 for sin θ.

cos2 θ = 15

16 Simplify.

cos θ = √15

4 Solve.

Now find sin 2θ.

sin 2θ = 2 sin θ cos θ Sine Double-Angle Identity

= 2(1

4) (

√15

4) sin θ =

1

4, cos θ =

√15

4

= √15

8 Simplify.

Exercises

1. If tan θ = 4

3 on the interval (𝜋,

3𝜋

2), find sin 2θ, cos 2θ, and tan 2θ.

Find the exact value of each expression.

2. sin 22.5° 3. cos 11𝜋

12

4. Solve cos θ sin 2θ = 0 on the interval [0, 2π).

𝟐𝟒

𝟐𝟓, –

𝟕

𝟐𝟓, –

𝟐𝟒

𝟕

√𝟐− √𝟐

𝟐 –

√𝟐+ √𝟑

𝟐

0, 𝝅

𝟐, 𝝅,

𝟑𝝅

𝟐

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Multiple-Angle and Product-to-Sum Identities

Use Product-to-Sum Identities

Product-to-Sum Identities Sum-to-Product Identities

sin α sin β = 1

2 [cos (α – β) – cos (α + β)]

cos α cos β = 1

2 [cos (α – β) + cos (α + β)]

sin α cos β = 1

2 [sin (α + β) + sin (α – β)]

cos α sin β = 1

2 [sin (α + β) – sin (α – β)]

sin α + sin β = 2 sin (𝛼 + 𝛽

2) cos (

𝛼 − 𝛽

2)

sin α – sin β = 2 cos (𝛼 + 𝛽

2) sin (

𝛼 − 𝛽

2)

cos α + cos β = 2 cos (𝛼 + 𝛽

2) cos (

𝛼 − 𝛽

2)

cos α – cos β = –2 sin (𝛼 + 𝛽

2) sin (

𝛼 − 𝛽

2)

Example: Rewrite sin 6x sin 4x as a sum or difference.

sin 6x sin 4x = 1

2 [cos (6x – 4x) – cos (6x + 4x)] Product-to-Sum Identity

= 1

2 (cos 2x – cos 10x) Simplify.

= 1

2 cos 2x –

1

2 cos 10x Distributive Property

Exercises

Rewrite each product as a sum or difference.

1. cos 3θ cos θ 2. sin 3x cos 5x

3. sin 7θ sin 4θ 4. sin 3θ cos 7θ

Find the exact value of each expression.

5. cos 3𝜋

4 + cos π 6. 6 cos 225° cos 135°

7. sin 105° – cos 15° 8. 2 sin 13𝜋

12 sin

7𝜋

12

Solve each equation.

9. cos 2θ – cos θ = 0 10. sin 2θ + sin 4θ = 0

11. cos 3θ + cos 2θ = 0 12. cos 4θ – cos 6θ = 0

𝟏

𝟐 cos 2θ +

𝟏

𝟐 cos 4θ

𝟏

𝟐 sin 8x –

𝟏

𝟐 sin 2x

𝟏

𝟐 cos 3θ –

𝟏

𝟐 cos 11θ

𝟏

𝟐 sin 10θ –

𝟏

𝟐 sin 4θ

–𝟐– √𝟐

𝟐 3

0 – 𝟏

𝟐

0 + 𝟐𝝅

𝟑n 0 +

𝝅

𝟑n,

𝝅

𝟐 + n𝝅

𝝅

𝟓 +

𝟐𝝅

𝟓n 0 +

𝝅

𝟓n

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


5-5 Practice Multiple-Angle and Product-to-Sum Identities

Find the values of sin 2θ, cos 2θ, and tan 2θ for the given value and interval.

1. sin θ = 12

13, (0°, 90°) 2. tan θ =

1

2, (𝜋,

3𝜋

2)

3. cos θ = 2

5, (−

𝜋

2, 0) 4. tan θ = – √3 , (

𝜋

2, 𝜋)

Solve each equation on the interval [0, 2π).

5. 2 sin θ cos θ = –1 6. 2 cos2 𝜃

2 – 3 cos θ = 0

Solve each equation.

7. cos 2θ + 2 cos2 θ = 0 8. sin 5θ + sin 7θ = 0

Rewrite each expression in terms with no power greater than 1.

9. cos4 𝜃

2 10. sin4 2θ

Write each product as a sum or difference.

11. cos 2θ cos θ 12. cos 5x sin 4x

13. BASEBALL A batter hits a ball with an initial velocity 𝑣0 of 100 feet per second at an angle θ to the horizontal. An outfielder catches the ball 200 feet from home plate. Find θ if the range of a projectile is given by the formula

R = 1

32 𝑣0

2 sin 2θ.

sin 2θ = 𝟏𝟐𝟎

𝟏𝟔𝟗; cos 2θ = –

𝟏𝟏𝟗

𝟏𝟔𝟗; sin 2θ =

𝟒

𝟓; cos 2θ =

𝟑

𝟓; tan 2θ =

𝟒

𝟑

tan 2θ = – 𝟏𝟐𝟎

𝟏𝟏𝟗

sin 2θ = – 𝟒√𝟐𝟏

𝟐𝟓; cos 2θ = –

𝟏𝟕

𝟐𝟓; sin 2θ = –

√𝟑

𝟐; cos 2θ = –

𝟏

𝟐;

tan 2θ = 𝟒√𝟐𝟏

𝟏𝟕 tan 2θ = √𝟑

𝟑𝝅

𝟒;

𝟕𝝅

𝟒

𝝅

𝟑;

𝟓𝝅

𝟑

𝝅

𝟑 + 𝝅n,

𝟐𝝅

𝟑 + 𝝅n 0 +

𝝅

𝟔 n

𝟑

𝟖 +

𝟏

𝟐 cos θ +

𝟏

𝟖 cos 2θ

𝟑

𝟖 –

𝟏

𝟐 cos 4θ +

𝟏

𝟖 cos 8θ

𝟏

𝟐 cos θ +

𝟏

𝟐 cos 3θ

𝟏

𝟐 sin 9x –

𝟏

𝟐 sin x

about 20˚

Documents

5-1 Study Guide and Intervention - MRS. FRUGE · 5-1 Study Guide and Intervention Trigonometric Identities Basic Trigonometric Identities An equation is an identity if the left side