Download Handout #9 - Essex County College

MTH 120 — Spring — 2008
Essex County College — Division of Mathematics
Handout — Version 91 — February 14, 2008
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Sum and Difference Identities
Using a unit circle (radius 1) and two angles A and B, where A > B. You should be able to
verify that the points along this circle corresponding to these two angles is: (cos A, sin A) and
(cos B, sin B). The distance between these two points is given by:
q
(cos A − cos B)2 + (sin A − sin B)2 .
If, so inclined, you were to simplify this you would find it is equivalent to:
p
2 − 2 (cos A cos B + sin A sin B).
Now if you were to take a new angle to be A − B, let’s call it S for the time being, it’s
point along the unit circle would be (cos S, sin S). The distance between this point and the
(1, 0) is identical to the distance between points (cos A, sin A) and (cos B, sin B). The distance
between(cos S, sin S) and (1, 0) is given by:
q
(cos S − 1)2 + (sin S − sin 0)2 .
Again, If, so inclined, you were to simplify this you would find it is equivalent to:
√
2 − 2 cos S.
Finally we have this relationship:
p
√
2 − 2 cos S =
2 − 2 (cos A cos B + sin A sin B)
2 − 2 cos S = 2 − 2 (cos A cos B + sin A sin B)
cos S = cos A cos B + sin AsinB
cos (A − B) = cos A cos B + sin A sin B
You need to know this identity:
cos (A − B) = cos A cos B + sin A sin B .
And from this identity we should be able to verify (need to know even odd relationships) this
identity:2
cos (A + B) = cos A cos B − sin A sin B .
1
2
This document was prepared by Ron Bannon using LATEX 2ε .
We’ll do this in class.
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Now using the cofunction identities we should be able to follow this argument.
π
sin θ = cos
−θ
h π2
i
sin (A + B) = cos
− (A + B)
h2 π
i
sin (A + B) = cos
−A −B
π
π2
− A cos B + sin
− A sin B
sin (A + B) = cos
2
2
sin (A + B) = sin A cos B + cos A sin B
You also need to know this identity:
sin (A + B) = sin A cos B + cos A sin B .
And from this identity we should be able to verify (need to know even odd relationships) this
identity:3
sin (A − B) = sin A cos B − cos A sin B .
Now, let’s verify the following identities.
1. sin 2x = 2 sin x cos x
2. cos 2x = cos2 x − sin2 x
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We’ll do this in class.
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3. cos 2x = 2 cos2 x − 1
4. cos 2x = 1 − 2 sin2 x
2
Examples
1. Find the exact value of cos (5π/12).
2. Find the exact value of sin 115◦ .
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3. Find the exact values of all six trigonometric functions for θ = 15◦
4. Given that sin α = 2/5 and sin β = 1/3, and that α and β are acute angles. Find each of
the following.
(a) sin (α + β) =
(b) sin (α − β) =
(c) cos (α + β) =
(d) cos (α − β) =
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(e) tan (α + β) =
(f) tan (α − β) =
(g) cot (α + β) =
(h) cot (α − β) =
(i) sec (α + β) =
(j) sec (α − β) =
(k) csc (α + β) =
(l) csc (α − β) =
5. Given that tan θ = −3/4 and θ is in the second quadrant, find each of the following.
(a) sin 2θ =
5
(b) cos 2θ =
(c) tan 2θ =
6. Verify the identity.
tan 2x =
2 tan x
1 − tan2 x
7. Verify the identity.
tan (A + B) =
6
tan A + tan B
1 − tan A tan B
8. Given that f (x) = sin x, show that
f (x + h) − f (x)
= sin x
h
cos h − 1
h
sin h
h
sin h
h
+ cos x
9. Given that f (x) = cos x, show that
f (x + h) − f (x)
= cos x
h
cos h − 1
h
10. Simplify.
1 + sin 2x + cos 2x
1 + sin 2x − cos 2x
11. Simplify.
sin 2x
2 cos2 x
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− sin x