Pre-Calculus Final - Chapter 6 - Trigonometric Identities
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)]}