Basic Trigonometric Identities and Equations

Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin2q + cos2q = 1 tan2q + 1 = sec2q cot2q + 1 = csc2q sin2q = 1 - cos2q tan2q = sec2q - 1 cot2q = csc2q - 1 cos2q = 1 - sin2q

Do you remember the Unit Circle? Where did our pythagorean identities come from?? Do you remember the Unit Circle? What is the equation for the unit circle? x2 + y2 = 1 What does x = ? What does y = ? (in terms of trig functions) sin2θ + cos2θ = 1 Pythagorean Identity!

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos2θ sin2θ + cos2θ = 1 . cos2θ cos2θ cos2θ tan2θ + 1 = sec2θ Quotient Identity Reciprocal Identity another Pythagorean Identity

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin2θ sin2θ + cos2θ = 1 . sin2θ sin2θ sin2θ 1 + cot2θ = csc2θ Quotient Identity Reciprocal Identity a third Pythagorean Identity

Using the identities you now know, find the trig value. 1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.

3.) sinθ = -1/3, find tanθ

Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. b) a)

Simplifing Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x d)

Simplify each expression.

Simplifying trig Identity Example: simplify tanxcosx sin x cos x tanx cosx tanxcosx = sin x

Simplifying trig Identity sec x csc x Example: simplify 1 cos x 1 cos x sinx = x sec x csc x 1 sin x = sin x cos x = tan x

Example Simplify: = cot x (csc2 x - 1) Factor out cot x = cot x (cot2 x) Use pythagorean identity = cot3 x Simplify

Example Simplify: = sin x (sin x) + cos x Use quotient identity cos x Simplify fraction with LCD = sin2 x + (cos x) cos x = sin2 x + cos2x cos x Simplify numerator = 1 cos x Use pythagorean identity = sec x Use reciprocal identity

*** Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity

Examples Prove tan(x) cos(x) = sin(x)

Examples Prove tan2(x) = sin2(x) cos-2(x)

Examples Prove

Examples Prove

Get common denominators Hints for Establishing Identities Get common denominators If you have squared functions look for Pythagorean Identities Work on the more complex side first If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities It's like a puzzle, you can use identities and algebra to get them to match!