Precalculus - Chapter 5

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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


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