Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 5 Trigonometric Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

5

Trigonometric Identities



5.1 Fundamental Identities

5.3 Sum and Difference Identities for Cosine

5.4 Sum and Difference Identities for Sine and Tangent

5.5Double-Angle Identities

5.6Half-Angle Identities

Trigonometric Identities5


Trigonometric Identities5.1Fundamental Identities ▪ Using the Fundamental Identities


If and is in quadrant IV, find each function value.

5.1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (page 191)

(a)

In quadrant IV, is negative.


If and is in quadrant IV, find each function value.

5.1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (cont.)

(b)

(c)


5.1 Example 2 Expressing One Function in Terms of Another

(page 192)


5.1 Example 3 Rewriting an Expression in Terms of Sine and Cosine (page 193)

Write in terms of and , and then simplify the expression so that no quotient appear.

2

2

1 tan

1 sec

2

2

1 tan

1 sec

2

2

1 tan

(sec 1)

2

2

sec

tan

2

2

2

1cossincos

2

1

sin 2csc


Sum and Difference Identities for Cosine5.3Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Find the exact value of each expression.

5.3 Example 1 Finding Exact Cosine Function Values

(page 206)

(a)


5.3 Example 1 Finding Exact Cosine Function Values

(cont.)

(b)

(c)


5.3 Example 2 Using Cofunction Identities to Find θ (page 208)

Find an angle θ that satisfies each of the following.


5.3 Example 3 Reducing cos (A – B) to a Function of a Single Variable (page 208)

Write cos(90° + θ) as a trigonometric function of θ alone.


5.3 Example 4 Finding cos (s + t) Given Information About s and t (page 209)

Suppose that , and both s and

t are in quadrant IV. Find cos(s – t).

The Pythagorean theorem gives

Since s is in quadrant IV, y = –8.


5.3 Example 4 Finding cos (s + t) Given Information About s and t (cont.)

Use a Pythagorean identity to find the value of cos t.

Since t is in quadrant IV,


5.3 Example 4 Finding cos (s + t) Given Information About s and t (cont.)


5.3 Example 5 Applying the Cosine Difference Identity to Voltage (page 221)

Because household current is supplied at different voltages in different countries, international travelers often carry electrical adapters to connect items they have brought from home to a power source. The voltage V in a typical European 220-volt outlet can be expressed by the function

(a) European generators rotate at precisely 50 cycles per second. Determine ω for these electric generators.

Each cycle is radians at 50 cycles per second.


5.3 Example 5 Applying the Cosine Difference Identity to Voltage (cont.)

(b) What is the maximum voltage in the outlet?

The maximum value of is 1.

The maximum voltage in the outlet is


5.3 Example 5 Applying the Cosine Difference Identity to Voltage (cont.)

(c) Determine the least positive value of in radians so that the graph of is the same as the graph of

Using the sum identity for cosine gives


Sum and Difference Identities for Sine and Tangent5.4Sum and Difference Identities for Sine ▪ Sum and Difference Identities for Tangent ▪ Applying the Sum and Difference Identities


Find the exact value of each expression.

5.4 Example 1 Finding Exact Sine and Tangent Function Values (page 217)

(a)


5.4 Example 1 Finding Exact Sine and Tangent Function Values (cont.)

(b)


5.4 Example 1 Finding Exact Sine and Tangent Function Values (cont.)

(c)


Write each function as an expression involving functions of θ.

5.4 Example 2 Writing Functions as Expressions Involving Functions of θ (page 218)

(a)

(b)

(c)


Suppose that A and B are angles in standard position

with and

Find each of the following.

5.4 Example 3 Finding Function Values and the Quadrant of A – B (page 218)

(c) the quadrant of A – B.


The identity for sin(A – B) requires sin A, cos A, sin B, and cos B. The identity for tan(A – B) requires tan A and tan B. We must find sin A, tan A, cos B and tan B.

5.4 Example 3 Finding Function Values and the Quadrant of A – B (cont.)

Because A is in quadrant III, sin A is negative and tan A is positive.

Because B is in quadrant IV, cos B is positive and tan B is negative.


5.4 Example 3(a) Finding Function Values and the Quadrant of A – B (cont.)

To find sin A and cos B, use the identity


5.4 Example 3(b) Finding Function Values and the Quadrant of A – B (cont.)

To find tan A and tan B, use the identity


5.4 Example 3(c) Finding Function Values and the Quadrant of A – B (cont.)

From parts (a) and (b), sin (A − B) < 0 and tan (A − B) < 0.

The only quadrant in which the values of both the sine and the tangent are negative is quadrant IV, so (A − B) is in quadrant IV.


Double-Angle Identities5.5Double-Angle Identities ▪ An Application ▪ Product-to-Sum and Sum-to-Product Identities


5.5 Example 1 Finding Function Values of 2θ Given Information About θ (page 225)

The identity for sin 2θ requires cos θ.


5.5 Example 1 Finding Function Values of 2θ Given Information About θ (cont.)


5.5 Example 1 Finding Function Values of 2θ Given Information About θ (cont.)

Alternatively, find tan θ and then use the tangent double-angle identity.


5.5 Example 2 Finding Function Values of θ Given Information About 2θ (page 226)

Find the values of the six trigonometric functions of θ

if

Use the identity to find sin θ:

θ is in quadrant III, so sin θ is negative.


5.5 Example 2 Finding Function Values of θ Given Information About 2θ (cont.)

Use the identity to find cos θ:

θ is in quadrant III, so cos θ is negative.


5.5 Example 2 Finding Function Values of θ Given Information About 2θ (cont.)


5.5 Example 4 Simplifying Expressions Using Double-Angle Identities (page 227)

Simplify each expression.


5.5 Example 5 Deriving a Multiple-Angle Identity (page 228)

Write cos 3x in terms of cos x.

Distributiveproperty.

Distributiveproperty.

Simplify.


Half-Angle Identities5.6Half-Angle Identities ▪ Applying the Half-Angle Identities


5.6 Example 1 Using a Half-Angle Identity to Find an Exact Value (page 234)

Find the exact value of sin 22.5° using the half-angle identity for sine.


5.6 Example 2 Using a Half-Angle Identity to Find an Exact Value (page 234)

Find the exact value of tan 75° using the identity


The angle associated with lies in quadrant II since

is positive while are negative.

5.6 Example 3 Finding Function Values of GivenInformation About s (page 234)

s2


5.6 Example 3 Finding Function Values of GivenInformation About s (cont.)

s2


5.6 Example 4 Simplifying Expressions Using the Half-Angle Identities (page 235)

Simplify each expression.

This matches part of the identity for .

Substitute 8x for A:

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 5 Trigonometric Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1