Trig Sum and Difference

Trigonometric Sum and Difference Theorems
1.
Prove the Basic Formula (The Formula for Cosine of a Difference)
(cosα, sin α) A
A
B (cosβ , sin β )
α
β
O (0, 0)
Consider any 2 points, A and B, on the unit circle. The radius segments to these points form angles
of measure α and measure β as shown in the figure. Point O(0, 0) is at the origin and the center of
the unit circle.
Since points A and B are on the unit circle, we know their coordinates to be:
A(cos α, sin α) and B(cos β, sin β)
In the triangle OB = OA = 1, since the radius of the unit circle is 1.
By the Distance Formula:
AB =
(cos α - cos β)2 + (sin α - sin β)2
(AB)2 = (cos α - cos β)2 + (sin α - sin β)2
(AB)2 = cos2 α - 2 cos α cos β + cos2 β + sin2 α - 2 sin α sin β + sin2 β
(AB)2 = 2 - 2(cos α cos β + sin α sin β)
By the Law of Cosines:
(AB)2 = (OA)2 + (OB)2 - 2(OA)(OB) cos(α - β)
(AB)2 = 1 + 1 - 2 • 1 • 1 • cos(α - β)
(AB)2 = 2 - 2 cos(α - β)
Since both of the above equations have (AB)2 on the left, then we can say:
2 - 2 cos(α - β) = 2 - 2(cos α cos β + sin α sin β)
Subtracting 2 from both sides, then dividing both sides by -2, we get:
cos(α - β) = cos α cos β + sin α sin β
Q.E.D.
2. Prove that cosine is an even
function:
ie cos(-x) = cos x
cos(-x) = cos(0 - x)
= cos 0 cos x + sin 0 sin x
= 1 * cos x + 0 * sin x
= cos x + 0
= cos x
Q.E.D.
3. Prove the formula regarding the cosine cos (π/2 - x) = cos (π/2) cos x + sin (π/2) sin x
of a compliment.
= 0 * cos x + 1 * sin x
= 0 + sin x
= sin x
Q.E.D.
4. Prove the formula regarding the sine
of a complement.
sin (π/2 - x) = cos [π/2 - (π/2 - x)]
= cos [(π/2 - π/2) + x]
= cos x
Q. E. D.
5. Prove that sine is an odd function.
sin (-x) = cos [π/2 - (-x)]
= cos (π/2 + x)
= cos (- π/2 - x)
= cos (- π/2) cos x + sin (- π/2) sin x
= 0 cos x - 1 sin x
= - sin x
Q. E. D.
6. Prove the formula for cosine of a
sum.
cos (x + y) = cos (x - -y)
= cos x cos (-y) + sin x sin (-y)
= cos x cos y - sin x sin y
Q. E. D.
7. Prove the formula for sine of a sum.
sin (x + y) = cos [π/2 - (x + y)]
= cos [(π/2 - x) - y]
= cos (π/2 - x) cos y + sin (π/2 - x) sin y
= sin x cos y + cos x sin y
Q. E. D.
8. Prove the formula for the sine of a
difference.
sin (x - y) = sin [x + (-y)]
= sin x cos (-y) + cos x sin (-y)
= sin x cos y - cos x sin y
Q. E. D.
9. Prove the formula for tangent of a
sum.
10 Prove the formula for tangent of a
. difference.
11 Prove that tangent is an odd
. function.
sin(x + y)
tan(x + y) = cos(x + y)
sin x cos y + cos x sin y
= cos x cos y - sin x sin y
sin x cos y + cos x sin y
cos x cos y
= cos x cos y - sin x sin y
cos x cos y
tan x + tan y
= 1 - tan x tan y
Q.E.D.
sin(x - y)
tan(x - y) = cos(x - y)
sin x cos y - cos x sin y
= cos x cos y + sin x sin y
sin x cos y - cos x sin y
cos x cos y
= cos x cos y + sin x sin y
cos x cos y
tan x - tan y
= 1 + tan x tan y
Q.E.D.
sin(-x)
tan(-x) = cos(-x)
- sin x
= cos x
= -tan x
Q.E.D.