The point of problems 1-6 was to
illustrate that the distributive
property does NOT hold for trig
functions.
If it does not hold, there must be
some other way to evaluate a
problem such as sin(3π/4 + 5π/6)
There are 6 more trig identities, called sum and
difference formulas that we will be using. You do
not need to memorize them:)
sin (u + v) = sinucosv + cosusinv
sin (u - v) = sinucosv - cosusinv
cos (u + v) = cosucosv - sinusinv
cos (u - v) = cosucosv + sinusinv
we will not
be using
tangent
identities
here, but
they do
exist
tan (u + v) =
tanu + tanv
1-tanutanv
tan (u - v) =
tanu - tanv
1+tanutanv
So in #9, how would you find the value of
sin(3π/4 + 5π/6)
use sin (u + v) = sinucosv + cosusinv
sin(3π/4)cos(5π/6) + cos (3π/4)sin(5π/6)
(√2/2) (-√3/2) + (-√2/2) (1/2)
-√6/4
+ -√2/4
-√6 - √2
4
Mar 2­8:02 AM
Mar 16­2:18 PM
3-16
a) find is cos 5π
6
b) find is cos π/2
c) find cos π/3
Mar 16­1:35 PM
Mar 16­1:58 PM
We use these formulas to help us evaluate trig functions that are not
on our mini trig tables:
Sum and difference formulas
Double angle formulas
sin (2x) = 2sinxcosx
cos (2x) = cos2x - sin2x
2cos2x - 1
1 - 2sin2x
12) ex. sin (π/12) is equivalent to sin (3π/4 - 2π/3)
to evaluate sin (π/12) we will use the difference formula for sin
sin (u - v) = sinucosv-cosusinv
sin (3π/4 - 2π/3) = sin(3π/4)cos(2π/3) - cos (3π/4)sin(2π/3)
= (√2/2)(-1/2)
(-√2/4)
- (-√2/2)(√3/2)
- (-√6/4)
-√2 + √6
4
What did you get for 11?
Mar 15­9:32 AM
Mar 2­8:12 AM
1
13) cos (45)cos(25) - sin(45)sin(25)
use cosucosv - sinusinv
14) sin(81)cos(22) - cos(81)sin(22)
16) prove cos (u - v) = cosucosv + sinusinv
use sinucosv-cosusinv
15) Prove:
sin(u-v) = sinucosv - cosusinv
(cos θ, sin θ )
sin(u-v)
=sin(u + -v)
= sinucos(-v) + cosusin(-v)
= sinucosv + cosu(-sinv)
= sinucosv - cosusinv
cos (u - v)
=cos (u + -v)
=cosucos(-v) - sinusin(-v)
=cosucosv - sinu(-sinv)
=cosucosv + sinusinv
θ
-θ
(cos -θ, sin -θ )
since cos is positive in
quad IV, and sin is
negative in quad IV,
cos (-θ) = cos θ
sin (-θ) = -sin θ
(these are called
opposite angle identities
- you will not need to
know them)
Mar 2­8:17 AM
Mar 2­8:24 AM
2