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5.3: Sum and Difference Identities
Transcript of 5.3: Sum and Difference Identities
(x + y) = x + y
f(x + y) = f(x) + f(y)
What's wrong with this:
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!
cos ( 60 ) = cos (90 - 30 )
= cos 90 cos 30 + sin 90 sin 30
= (0)(.866) + (1) (.5)
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
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?
Use sum/difference identities to find the exact value:
1. sin 15
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. Simplify to either 1 or -1:
sec (-x) - tan x
3. Find all solutions on the interval [0, 2(pi) ).
(No calculator is necessary.)
sin x tan x = sin x