Chapter 5 Precalc

Methods of Verifying Trigonometric Identities

1. Start with the more complicated side and transform it to the simpler side 2. Stay focused of the final expression 3. Convert to sines and cosines 4. Work on both sides 5. Use conjugates

trigonometric equation

An equation with a variable in place of the value of an angle.

Reduction Formula

If α is an angle in standard position whose terminal side contains (a,b), then for any real number x: a sin x + b cos x = √(a² + b²) sin(x + α)

Fundamental trigonometric identities

Reciprocal Identities, Quotient Identities, Pythagorean Identities, Even-Odd Identities

extraneous solution

an apparent solution that must be rejected because it does not satisfy the original equation

Sum Formula for Cosine

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

Difference Formula for Cosine

cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

multiple angles of x

if x is the measure of an angle, then for any real number k, the number is kx

Cofunction Identities

sin (π/2 - x) = cos x cos (π/2 - x) = sin x tan (π/2 - x) = cot x cot (π/2 - x) = tan x sec (π/2 - x) = csc x csc (π/2 - x) = sec x

Sum to Product Formulas

sin x + sin y = 2sin(x + y)/(2) cos(x -y)/(2) sin x - sin y = 2cos(x + y)/(2) sin (x - y)/(2) cos x + cos y = 2cos(x + y)/(2) cos(x - y)/(2) cos x - cos y = -2 sin(x + y)/(2) sin(x - y)/(2)

Half Angle Formulas

sin x/2 = + - √(1 - cos x)/2 cos x/2 = + - √1 + cos x)/2 tan x/2 = + -√(1 - cos x)/(sin x)= (sin x)/(1 + cos x)=1-cosx/(sinx)

Even/Odd Identities

sin(-x) = - sin x cos(-x) = cos x tan (-x) = - tan x csc (-x) = - csc x sec (-x) = sec x cot (-x) = - cot x

Double Angle Formulas

sin(2x)=2sin(x)cos(x) cos(2x)=cos^2-sin^2 cos(2x)=1-2sin^2 cos(2x)=2cos^2-1 tan(2x)=2tanx/1-tan^2

Sum Formula for Sine

sin(a+b) = sinacosb + cosasinb

Difference Formula for Sine

sin(a-b) = sin(a)cos(b)-cos(a)sin(b)

Power-Reducing Formulas

sin^2 x = (1 - cos 2x)/(2) cos^2 x = (1 + cos 2x)/(2) tan^2 x = (1 - cos 2x)/(1 + cos 2x)

Pythagorean Identities

sin^2x+cos^2x=1 1+tan^2x=sec^2x 1+cot^2x=csc^2x

Product to Sum Formulas

sinx sin y = 1/2 [cos(x - y) - cos(x + y)] cosx cos y = 1/2 [cos(x - y) + cos(x + y)] sinx cos y = 1/2 [sin(x + y) + sin(x - y)] cos x sin y = 1/2 [sin(x + y) - sin(x - y)]

Reciprocal Identities

sinθ = 1/cscθ ; cscθ = 1/sinθ cosθ = 1/secθ ; secθ = 1/cosθ tanθ = 1/cotθ ; cotθ = 1/tanθ

Sum Formula for Tangent

tan(a+b)=tan(a)+tan(b)/1-tan(a)tan(b)

Difference Formula for Tangent

tan(a-b) = (tan(a) - tan(b)) / 1 + tan(a)tan(b)

Quotient Identities

tanθ = sinθ/cosθ cotθ = cosθ/sinθ

Trigonometric Substitution

to convert algebraic expressions to trigonometric expressions

verifying trigonometric identities

transforming one side of the equation into the other side by sequence of steps, each of which produces an identity. The steps involved can be algebraic manipulations or can use known identities.

simplifying Trigonometric Expressions

use the inverse function definitions along with the fundamental trigonometric identity

cos x = cos a

x = a + 2nπ or x = (2π- a) + 2nπ

sin x = sin a

x = a + 2nπ or x = (π - a) + 2nπ

tan x = tan a

x = a nπ