Slide 1-1 5.1 Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference...

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5.1 Fundamental Identities

5.2 Verifying Trigonometric Identities

5.3 Sum and Difference Identities for Cosine

5.4 Sum and Difference Identities for Sine and Tangent

5.5 Double-Angle Identities

5.6 Half-Angle Identities

Chapter 5

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Formulas and Identities

Negative Angle Identities

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Formulas and Identities

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Formulas and Identities

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Fundamental Identities5.1Fundamental Identities ▪ Using the Fundamental Identities

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If and θ is in quadrant II, find each

function value.

Example FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT

(a) sec θ

In quadrant II, sec θ is negative, so

Pythagorean identity

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(b) sin θ

from part (a)

Quotient identity

Reciprocal identity

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(b) cot(– θ) Reciprocal identity

Negative-angle identity

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Write cos x in terms of tan x.

Example EXPRESSING ONE FUNCITON IN TERMS OF ANOTHER

Since sec x is related to both cos x and tan x by identities, start with

Take reciprocals.

Reciprocal identity

Take the square root of each side.

The sign depends on the quadrant of x.

2

2

1 tancos

1 tan

xx

x

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Write in terms of sin θ and cos θ, and

then simplify the expression so that no quotients appear.

2

2 2

2

2

cos11 cot sin

11 csc 1sin

Quotient identities

Multiply numerator and denominator by the LCD.

2

2

1 cot

1 csc

22

2

22

cos1 sin

sin

11 sin

sin

Example

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Reciprocal identity

Pythagorean identities

2 2

2

sin cos

sin 1

2

1

cos

2sec

Distributive property

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Verifying Trigonometric Identities

5.2

Strategies ▪ Verifying Identities by Working With One Side ▪ Verifying Identities by Working With Both Sides

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As you select substitutions, keep in mind the side you are not changing, because it represents your goal.For example, to verify the identity

find an identity that relates tan x to cos x.

Since andthe secant function is the best link between the two sides.

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Hints for Verifying Identities

If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin2 x, which could be replaced with cos2 x. Similar procedures apply for 1 – sin x,

1 + cos x, and 1 – cos x.

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Verifying Identities by Working with One Side

To avoid the temptation to use algebraic properties of equations to verify identities, one strategy is to work with only one side and rewrite it to match the other side.

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Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)

Verify that the following equation is an identity.

Work with the right side since it is more complicated.Right side of

given equation

Distributive property

Left side of given equation

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Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)

Verify that the following equation is an identity.

Distributive property

Left side

Right side

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Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)

Verify that is an identity.

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VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)

Verify that is an identity.

Multiply by 1 in the form

Example

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Verifying Identities by Working with Both Sides

If both sides of an identity appear to be equally complex, the identity can be verified by working independently on each side until they are changed into a common third result.

Each step, on each side, must be reversible.

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Example VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES)

Verify that is an identity.

Working with the left side: Multiply by 1 in the form

Distributive property

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Working with the right side:

Factor the numerator.

Factor the denominator.

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So, the identity is verified.

Left side of given equation

Right side of given equation

Common third expression

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Sum and Difference Identities for Cosine

5.3

Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities ▪ Verifying an Identity

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Example FINDING EXACT COSINE FUNCTION VALUES

Find the exact value of cos 15 .

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Example FINDING EXACT COSINE FUNCTION VALUES

Find the exact value of

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Example FINDING EXACT COSINE FUNCTION VALUES

Find the exact value of cos 87cos 93 – sin 87sin 93 .

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Example USING COFUNCTION IDENTITIES TO FIND θ

Find one value of θ or x that satisfies each of the following.(a) cot θ = tan 25°

(b) sin θ = cos (–30°)

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Example USING COFUNCTION IDENTITIES TO FIND θ (continued)

(c)

Find one value of θ or x that satisfies the following.

3csc

4sec x

3csc sec

4x

3csc

4csc

2x

3

4 2x

4x

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Example REDUCING cos (A – B) TO A FUNCTION OF A SINGLE VARIABLE

Write cos(180° – θ) as a trigonometric function of θ alone.

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Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t

Suppose that and both s and

t are in quadrant II. Find cos(s + t).

Sketch an angle s in quadrant II such that Since let y = 3 and r =5.

The Pythagorean theorem gives

Since s is in quadrant II, x = –4 and

Method1

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Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.)

Sketch an angle t in quadrant II such that Since let x = –12 and r = 13.The Pythagorean theorem gives

Since t is in quadrant II, y = 5 and

12cos ,

13

xt

r

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Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.)

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Example

Method2We use Pythagorean identities here. To find cos s, recall that sin2s + cos2s = 1, where s is in quadrant II. 2

23cos 1

5s

29

cos 125

s

2 16cos

25s

4cos

5s

sin s = 3/5

Square.

Subtract 9/25

cos s < 0 because s is in quadrant II.

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Example

To find sin t, we use sin2t + cos2t = 1, where t is in quadrant II.

22 12

sin 113

t

2 144sin 1

169t

2 25sin

169t

5sin

13t

cos t = –12/13

Square.

Subtract 144/169

sin t > 0 because t is in quadrant II.

From this point, the problem is solved using (see Method 1).

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Sum and Difference Identities for Sine and Tangent

5.4

Sum and Difference Identities for Sine ▪ Sum and Difference Identities for Tangent ▪ Applying the Sum and Difference Identities ▪ Verifying an Identity

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Sum and Difference Identities for Tangent

Fundamental identity

Sum identities

Multiply numerator and denominator by 1.

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Sum and Difference Identities for Tangent

Multiply.

Simplify.

Fundamental identity

Replace B with –B and use the fact that tan(–B) = –tan B to obtain the identity for the tangent of the difference of two angles.

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Tangent of a Sum or Difference

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Example FINDING EXACT SINE AND TANGENT FUNCTION VALUES

Find the exact value of sin 75 .

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Example FINDING EXACT SINE AND TANGENT FUNCTION VALUES

Find the exact value of

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Example FINDING EXACT SINE AND TANGENT FUNCTION VALUES

Find the exact value of

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Example WRITING FUNCTIONS AS EXPRESSIONS INVOLVING FUNCTIONS OF θ

Write each function as an expression involving functions of θ.

(a)

(b)

(c)

cos 3 sin

2

sin(180 ) sin180 cos cos180 sin 0 cos ( 1)sin

sin

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Example FINDING FUNCTION VALUES AND THE QUADRANT OF A + B

Suppose that A and B are angles in standard position withFind each of the following.

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

Because A is in quadrant II, cos A is negative and tan A is negative.

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

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(a)

(b)

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(c) 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 positive is quadrant I, so (A + B) is in quadrant I.

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Example VERIFYING AN IDENTITY USING SUM AND DIFFERENCE IDENTITIES

Verify that the equation is an identity.

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

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Half-Angle Identities5.6Half-Angle Identities ▪ Applying the Half-Angle Identities ▪ Verifying an Identity

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Half-Angle Identities

We can use the cosine sum identities to derive half-angle identities.

Choose the appropriate sign depending on the quadrant of

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Half-Angle Identities

2 1 cos2cos

2

xx

1 cos2cos

2

xx

Choose the appropriate sign depending on the quadrant of

1 coscos

2 2

A A

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Half-Angle Identities

There are three alternative forms for

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Half-Angle Identities

sintan ,

2 1 cos

A A

A

From the identity we can also derive an equivalent identity.

1 costan

2 sin

A A

A

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Half-Angle Identities

1 coscos

2 2

A A

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Example USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUE

Find the exact value of cos 15° using the half-angle identity for cosine.

Choose the positive square root.

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Example USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUE

Find the exact value of tan 22.5° using the identity

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Example FINDING FUNCTION VALUES OF s/2 GIVEN INFORMATION ABOUT s

The angle associated with lies in quadrant II

since

is positive while are negative.

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Example SIMPLIFYING EXPRESSIONS USING THE HALF-ANGLE IDENTITIES

Simplify each expression.

Substitute 12x for A:

This matches part of the identity for

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Example VERIFYING AN IDENTITY

Verify that is an identity.

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Double-Angle Identities

We can use the cosine sum identity to derive double-angle identities for cosine.

Cosine sum identity

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Double-Angle Identities

There are two alternate forms of this identity.

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Double-Angle Identities

We can use the sine sum identity to derive a double-angle identity for sine.

Sine sum identity

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Double-Angle Identities

We can use the tangent sum identity to derive a double-angle identity for tangent.

Tangent sum identity

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Double-Angle Identities

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Example FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ

Given and sin θ < 0, find sin 2θ, cos 2θ, and tan 2θ.To find sin 2θ, we must first find the value of sin θ. 2

2 3sin 1

5

2 16sin

25

4sin

5

Now use the double-angle identity for sine.4 3 24

sin2 2sin cos 25 5 25

Now find cos2θ, using the first double-angle identity for cosine (any of the three forms may be used).

2 2 9 16 7cos2 cos sin

25 25 25

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

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Alternatively, find tan 2θ by finding the quotient of sin 2θ and cos 2θ.

24sin2 2425tan2

7cos2 725

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Example FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ

Find the values of the six trigonometric functions of θ if

We must obtain a trigonometric function value of θ alone.

θ is in quadrant II, so sin θ is positive.

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Use a right triangle in quadrant II to find the values of cos θ and tan θ.

Use the Pythagorean theorem to find x.

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Example VERIFYING A DOUBLE-ANGLE IDENTITY

Quotient identity

Verify that is an identity.

Double-angle identity

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Example SIMPLIFYING EXPRESSIONS USING DOUBLE-ANGLE IDENTITIES

Simplify each expression.

Multiply by 1.

cos2A = cos2A – sin2A

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Example DERIVING A MULTIPLE-ANGLE IDENTITY

Write sin 3x in terms of sin x.

Sine sum identity

Double-angle identities

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Product-to-Sum Identities

We can add the identities for cos(A + B) and cos(A – B) to derive a product-to-sum identity for cosines.

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Product-to-Sum Identities

Similarly, subtracting cos(A + B) from cos(A – B) gives a product-to-sum identity for sines.

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Product-to-Sum Identities

Using the identities for sin(A + B) and sin(A – B) in the same way, we obtain two more identities.

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Product-to-Sum Identities

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Example

Write 4 cos 75° sin 25° as the sum or difference of two functions.

USING A PRODUCT-TO-SUM IDENTITY

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Sum-to-Product Identities

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Example

Write as a product of two functions.

USING A SUM-TO-PRODUCT IDENTITY