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The derivation of the sum and difference formulas for cosine and sine, as well as the double-angle formulas. It covers the difference formula for cosine, the sum formula for cosine, and the difference and sum formulas for sine. The document also includes the even-odd identities and some examples.
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Difference Formula for Cosine
Consider the following two diagrams:
P = (cos v, sin v)
Q = (cos u, sin u) u v
u − v
x^2 + y^2 = 1
u − v
B = (cos(u − v), sin(u − v))
x^2 + y^2 = 1
The triangles P OQ and AOB are congruent. Therefore, d(P, Q) = d(A, B).
By squaring both sides, we have that
d(P, Q)^2 = d(A, B)^2.
We calculate both sides separately:
d(P, Q)^2 = (cos u − cos v)^2 + (sin u − sin v)^2
= cos^2 u − 2 cos u cos v + cos^2 v + sin^2 u − 2 sin u sin v + sin^2 v Expand the squares = (cos^2 u + sin^2 u) + (cos^2 v + sin^2 v) − 2 cos u cos v − 2 sin u sin v Group terms = 1 + 1 − 2 cos u cos v − 2 sin u sin v Pythagorean identity = 2 − 2 cos u cos v − 2 sin u sin v. d(A, B)^2 = (cos(u − v) − 1)^2 + (sin(u − v) − 0)^2 = cos^2 (u − v) − 2 cos(u − v) + 1 + sin^2 (u − v) Expand the squares = 1 − 2 cos(u − v) + 1 Pythagorean identity = 2 − 2 cos(u − v).
Therefore,
d(A, B)^2 = d(P, Q)^2 =⇒ 2 − 2 cos(u − v) = 2 − 2 cos u cos v − 2 sin u sin v =⇒ −2 cos(u − v) = −2 cos u cos v − 2 sin u sin v Subtract 2 on both sides =⇒ cos(u − v) = cos u cos v + sin u sin v Divide by − 2 on both sides.
Difference Formula for Cosine Therefore, the difference formula for cosine is cos(u − v) = cos u cos v + sin u sin v. Sum Formula for Cosine cos(u + v) = cos(u − (−v)) = cos u cos(−v) + sin u sin(−v) Difference formula = cos u cos v − sin u sin v Even-Odd Identities. Therefore, the sum formula for cosine is cos(u + v) = cos u cos v − sin u sin v.
Summary of Identities i) Reciprocal Identities csc x = (^) sin^1 x sec x = (^) cos^1 x cot x = (^) tan^1 x ii) Quotient Identities tan x = cossin^ xx cot x = cos sin xx iii) Pythagorean Identities sin^2 x + cos^2 x = 1 1 + tan^2 x = sec^2 x 1 + cot^2 x = csc^2 x iv) Even-Odd Identities sin(−x) = − sin x cos(−x) = cos x tan(−x) = − tan x v) Sum/Difference Formulas cos(u − v) = cos u cos v + sin u sin v cos(u + v) = cos u cos v − sin u sin v sin(u − v) = sin u cos v − cos u sin v sin(u + v) = sin u cos v + cos u sin v vi) Double-Angle Formulas sin(2u) = 2 sin u cos u cos(2u) = cos^2 u − sin^2 u = 2 cos^2 u − 1 tan(2u) = (^1) −2 tan tan^ u (^2) u cos(2u) = cos^2 u − sin^2 u = 1 − 2 sin^2 u
Example: Find the exact value of
cos
(π 3 −^
π 4
Example: Find the exact value of sin
( 5 π 12
Example: Given that tan u = 5/12, with u in quadrant I, and cos v = 3/5, with v in quadrant IV, find sin(u − v) and cos(u + v).
A trigonometric equation is an equation involving one or more trigonometric func- tions. Like verifying trigonometric identites, there is no one method that will work for all problems, but usually a good strategy is to try to use trigonometric identities to simplify the equation, and then solve the equation using algebraic techniques. Since trigonometric functions are periodic, trigonometric equations have an infinite number of solutions; however, we will usually restrict our search to solutions in the interval [0, 2 π).
Example: Find all solutions in the interval [0, 2 π) of the following equations:
i) 2 sin x − √2 = 0
ii) sin x + 1 = sin(−x)
iii) 3 tan^2 x − 1 = 0
iv) 2 sin(3x) − √2 = 0
v) cos(2x) = sin x
