**5.3: Sum and Difference Identities**

(x + y) = x + y

f(x + y) = f(x) + f(y)

What's wrong with this:

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2

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This means you can't assume that

sin (x + y) = sin x + sin y, or

cos (x + y) = cos x + cos y...

because it DOESN'T!!

Cosine of a sum or difference

cos (u + v) = cos u cos v + sin u sin v

NOTE: sign changes!

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Example:

cos ( 60 ) = cos (90 - 30 )

= cos 90 cos 30 + sin 90 sin 30

= (0)(.866) + (1) (.5)

= .5

o

o

o

o

o

o

o

Why does this work??

Sine of a sum or difference

sin (u + v) = sin u cos v + cos u sin v

NOTE: sign does NOT change

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Ex:

sin (3(pi)/4) = sin ((pi)/2 + (pi)/4)

= sin (pi/2) cos (pi/4) + cos (pi/2) sin (pi/4)

= (0) (sqrt(2)/2) + (1) (sqrt(2)/2

= sqrt(2) / 2

How would you do the TANGENT of a difference or sum?

Examples:

Use sum/difference identities to find the exact value:

1. sin 15

o

Write the expression as the sine, cosine, or tangent of an angle:

2. sin 3x cos x - cos 3x sin x

Prove the identity:

3. sin (x - (pi)/2) = -cos x

4. cot ( (pi)/2 - u) = tan u

Express the function as a sinusoid in the form

y = a sin (bx + c).

5. y = 3 sin x + 4 cos x

**Assignment Quiz: 5.1**

**1. Use basic identities to simplify:**

______________

______________

**1 - cos u**

2

**sin u**

**2. Simplify to either 1 or -1:**

sec (-x) - tan x

sec (-x) - tan x

2

2

**3. Find all solutions on the interval [0, 2(pi) ).**

(No calculator is necessary.)

sin x tan x = sin x

(No calculator is necessary.)

sin x tan x = sin x

2