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
