### Present Remotely

Send the link below via email or IM

Present to your audience

• Invited audience members will follow you as you navigate and present
• People invited to a presentation do not need a Prezi account
• This link expires 10 minutes after you close the presentation
• A maximum of 30 users can follow your presentation

Do you really want to delete this prezi?

Neither you, nor the coeditors you shared it with will be able to recover it again.

# 5.3: Sum and Difference Identities

Precalculus
by

## Jessica Edrington

on 4 March 2015

Report abuse

#### Transcript of 5.3: Sum and Difference Identities

5.3: Sum and Difference Identities
(x + y) = x + y

f(x + y) = f(x) + f(y)
What's wrong with this:
2
2
2
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!
_
_
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
_
_
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

2
2
3. Find all solutions on the interval [0, 2(pi) ).
(No calculator is necessary.)
sin x tan x = sin x

2
Full transcript