Basic Trigonometric Identities and Equations By the end of this chapter, you should be able to: -identify non-permissible values for trigonometric expressions -show that a trigonometric identity is true for all permissible values of the variable by using algebra (not just by substituting numbers in for the variable or by graphing) -Use trigonometric identities to simplify more complicated trigonometric expressions -solve trigonometric equations algebraically -find exact values for given trigonometric expressions

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 5.4.3

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.

Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. b) a) 5.4.5

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

Simplify each expression.

Simplifying trig Identity Example1: simplify tanxcosx sin x cos x tanx cosx tanxcosx = sin x

Simplifying trig Identity sec x csc x Example2: simplify 1 cos x 1 cos x sinx = x sec x csc x 1 sin x = sin x cos x = tan x

Simplifying trig Identity cos2x - sin2x cos x Example2: simplify = sec x cos2x - sin2x cos x cos2x - sin2x 1

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

Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity

One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this: substitute using each identity simplify

Another way to use identities is to write one function in terms of another function. Let’s see an example of this: This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

Sum and Difference Identities

Combined Sum and Difference Formulas

These identities are useful to find exact answers for non-special angles Example Find the exact value of the following. cos 15° cos (or 60° – 45°)

Example Find the exact value of the following. sin 75° tan sin 40° cos 160° – cos 40° sin 160° Solution (a)

(b) (c) sin 40°cos 160° – cos 40°sin 160° =sin(40°-160°) = sin(–120°)

Example Find the exact value of ( cos 80° cos 20° + sin 80° sin 20°) . Solution The given expression is the right side of the formula for cos( - ) with  = 80° and  = 20°. cos( -) = cos  cos  + sin  sin  cos 80° cos 20° + sin 80° sin 20° = cos (80° - 20°) = cos 60° = 1/2

Example Write the following expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. Solution:

DOUBLE-ANGLE IDENTITIES

If we want to know a formula for we could use the sum formula. we can trade these places This is called the double angle formula for sine since it tells you the sine of double 

Let's try the same thing for This is the double angle formula for cosine but by substiuting some identities we can express it in a couple other ways.

Double-angle Formula for Tangent

Summary of Double-Angle Formulas

Your Turn: Simplify an Expression Simplify cot x cos x + sin x. Click for answer. Page 189

Your Turn: Cosine Sum and Difference Identities Find the exact value of cos 75°. Click for answer. Page 198

Your Turn: Sine Sum and Difference Identities Find the exact value of . Click for answer.

Your Turn: Double-Angle Identities If , find sin 2x given sin x < 0. Click for answer.

Your Turn: Double-Angle Identities