Pre-Calculus Final - Chapter 6 - Trigonometric Identities

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Even/Odd Identity: cot(-θ) =

-cot(θ)

Even/Odd Identity: csc(-θ) =

-csc(θ)

Even/Odd Identity: sin(-θ) =

-sin(θ)

Even/Odd Identity: tan(-θ) =

-tan(θ)

cos a = (To find sin or to find cos from sin.):

-√1-sin²a

pythagorean identity: sin²(θ) + cos²(θ) =

1

Establish the Identity (steps)

1. Stat with the more complicated side. 2. Rewrite sums or differences of quotients of a single quotient. 3. Rewrite one side in terms of sines and cosines. 4. Keep goal in mind. 5. Manipulate only one side.

Techniques to simplify equations:

1. Write in terms of sin and cosin. 2. Multiply by conjugate. 3. Find a common denominator. 4. Factor Expression.

Cos Quadrants

1st and 2nd

double-angle identity: sin(2θ) =

2sin(θ)cos(θ)

double-angle identity: tan(2θ) =

2tan(θ) ÷ [1 − tan²(θ)]

Sin/Tan Quadrants

4st and 1st

If not an Identity it is a...

Conditional Equation

Cos In terms of y, x, and r:

Cos = x/r

If f(x) = g(x)

Identity

Sin In terms of y, x, and r:

Sin = y/r

Tan(θ) × Cos(θ) =

Sin(θ)

Tan In terms of y, x, and r:

Tan = y/x

angle-sum identity: tan(α+β) = (either odd)

[sin(α+β)] ÷ [cos(α+β)]

angle-difference identity: tan(α−β) = (either odd)

[sin(α-β)] ÷ [cos(α-β)]

angle-sum identity: tan(α+β) = (both angles not)

[tan(α) + tan(β)] ÷ [1 − tan(α)tan(β)]

angle-difference identity: tan(α−β) = (both angles not)

[tan(α) − tan(β)] ÷ [1 + tan(α)tan(β)]

angle-difference identity: cos(α−β) =

cos(α)cos(β) + sin(α)sin(β)

angle-sum identity: cos(α+β) =

cos(α)cos(β) − sin(α)sin(β)

Even/Odd Identity: cos(-θ) =

cos(θ)

co-function identity: sin[(π÷2) − θ] =

cos(θ)

double-angle identity: cos(2θ) =

cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 2sin²(θ) − 1

co-function identity: tan[(π÷2) − θ] =

cot(θ)

pythagorean identity: cot²(θ) + 1 =

csc²(θ)

Even/Odd Identity: sec(-θ) =

sec(θ)

pythagorean identity: tan²(θ) +1 =

sec²(θ)

angle-sum identity: sin(α+β) =

sin(α)cos(β) + cos(α)sin(β)

angle-difference identity: sin(α−β) =

sin(α)cos(β) − cos(α)sin(β)

co-function identity: cos[(π÷2) − θ] =

sin(θ)

Sin x= 0 only if

x=kπ

Sin/Cos (In terms of k)

±2kπ

Tan (In terms of k)

±kπ

half-angle identity: cos(π/2) =

±√{[1 + cos(θ)] ÷ 2}

half-angle identity: sin(π/2) =

±√{[1 − cos(θ)] ÷ 2}

half-angle identity: tan(π/2) =

±√{[1 − cos(θ)] ÷ [1 + cos(θ)]}


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