Objective Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities.

Pythagorean Theorem x2 + y2 = r2 Note: This will also be the equation for the unit circle

A derivation for a Pythagorean identity is shown below. x2 + y2 = r2 Pythagorean Theorem Divide both sides by r2. Substitute cos θ for and sin θ for cos2 θ + sin2 θ = 1

y r x

Reciprocal Identities

Quotient Identities y x = x y =

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 1st Pythagorean Identity!

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos2θ sin2θ + cos2θ = 1 sin2θ + cos2θ = 1 . cos2θ cos2θ cos2θ tan2θ + 1 = sec2θ Quotient Identity Reciprocal Identity 2nd Pythagorean Identity

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin2θ sin2θ + cos2θ = 1 sin2θ + cos2θ = 1 . sin2θ sin2θ sin2θ 1 + cot2θ = csc2θ Quotient Identity Reciprocal Identity 3rd Pythagorean Identity

Opposite Angle Identities sometimes these are called even/odd identities

Other versions of the Pythagorean identities may be used also. Look at and solve for cos2 θ cos2 θ + sin2 θ = 1 This should leave you with: cos2 θ = 1 - sin2 θ Do this again this time solving for sin2 θ. This should leave you with: sin2 θ = 1 - cos2 θ

If cot θ = 2 and cos θ < 0, find sin θ and cos θ. Use Pythagorean Identities If cot θ = 2 and cos θ < 0, find sin θ and cos θ. Use the Pythagorean Identity that involves cot θ. cot 2 θ + 1 = csc 2 θ Pythagorean Identity (2) 2 + 1 = csc 2 θ cot θ = 2 5 = csc 2 θ Simplify. = csc θ Take the square root of each side. Reciprocal Identity Solve for sin θ.

You may start with either side of the given equation You may start with either side of the given equation. It is often easier to begin with the more complicated side and simplify it to match the simpler side. Helpful Hint If you get stuck, try converting all of the trigonometric functions to sine and cosine functions.

Other strategies you may need to use are: Get a common denominator. Split up a fraction that has a monomial in the denominator. Distribute. Simplify complex fractions.