Precalculus - Chapter 5
cos θ/2
+-√1+cosθ/2
tan θ/2
+-√1-cosθ/1+cosθ =sin θ/1+cosθ =1 - cosθ/sinθ
sin θ/2
+-√1-cosθ/2
cos x - cos y
-2 sin(x+y /2) sin(x-y /2)
5 techniques for verifying identities
-Simplify the more complicated side -Focus on the final expression -Convert to sines and cosines -Transform both sides to the same expression -Multiply by conjugates
cos^2 x
1 + cos 2x / 2
tan^2 x
1 - cos 2x / 1 + cos 2x
sin^2 x
1 - cos 2x / 2
cos x cos y
1/2 [cos(x-y) + cos(x+y)]
sin x sin y
1/2 [cos(x-y) - cos(x+y)]
sin x cos y
1/2 [sin(x+y) + sin(x-y)]
cos x sin y
1/2 [sin(x+y) - sin(x-y)]
sec x
1/cos x
tan x (reciprocal)
1/cot x
sin x
1/csc x
cos x
1/sec x
csc x
1/sin x
cot x (reciprocal)
1/tan x
cos x + cos y
2 cos(x+y /2) cos(x-y /2)
sin 2x
2 sin x cos x
sin x + sin y
2 sin(x+y /2) cos(x-y /2)
sin x - sin y
2 sin(x-y /2) cos (x+y /2)
tan 2x
2 tan x / 1 - tan^2 x
sin^2x+cos^2x
=1
Reduction Formula' a sin x + b cos x =
A sin(x + θ) A=√a^2 + b^2
Which functions are even? f(-x)=f(x)
Cosine and Secant
Which functions are odd? f(-x)=-f(x)
Sine, Cosecant, Tangent, and Cotangent
cos(u-v)
cos u cos v + sin u sin v
cos(u+v)
cos u cos v - sin u sin v
sin(π/2 - v)
cos v
cot x (quotient)
cos x/sin x
cos 2x
cos^2 x - sin^2 x 1 - 2 sin^2 x 2 cos^2 x -1
tan(π/2 - v)
cot v
sec(π/2 - v)
csc v
1 + cot^2x
csc^2 x
csc(π/2 - v)
sec v
1+tan^2x
sec^2x
sin(u + v)
sin u cos v + cos u sin v
sin(u - v)
sin u cos v - cos u sin v
cos(π/2 - v)
sin v
tan x (quotient)
sin x/cos x
tan(u + v)
tan u + tan v / 1 - tan u tan v
tan(u - v)
tan u - tan v / 1 + tan u tan v
cot(π/2 - v)
tan v
To verify a trigonometric identity, you must...
use formulas/known identities and algebra to make one side equal the other (or transform both sides to the same thing)
How do you find all solutions of cos x=cos α?
x = α + 2nπ x = (2π - α) + 2nπ
How do you find all solutions of sin x=sin α?
x = α + 2nπ x = (π - α) + 2nπ
How do you find all solutions of tan x = tan α?
x = α + nπ
