Trigonometric Identity Review

Trigonometry Identities Reciprocal Identities sin θ = cos θ = tan θ = Quotient Identities Tan θ = cot θ =

Pythagorean Identities Try to come up with an identity using the pythagorean theorem (a 2 + b 2 = c 2 ) and the unit circle

NOW divide your identity with sin 2 x and then cos 2 x and come up with two more identities: sin 2 x + cos 2 x = 1

Pythagorean Identities cos 2 θ + Sin 2 θ = tan 2 θ = sec 2 θ cot 2 θ + 1 = csc 2 θ

Using Identities Find sin θ and cos θ if tan θ = 5 and cos θ > 0

Cofunction Identities ANGLE A:ANGLE B: sinA = y/rsinB = CosA = x/rCosB = tanA = y/xtanB = cscA = r/ycscB = secA = r/xsecB = cotA = x/ycotB = A B

Confunction Identities Sin( π /2 – θ ) = cos θ Cos( π /2 – θ ) = sin θ Tan( π /2 – θ ) = cot θ Csc( π /2 – θ ) = sec θ Sec( π /2 – θ ) = csc θ Cot( π /2 – θ ) = tan θ

Odd-Even Identities Sin(-x) = -sinx Cos(-x) = _______ Tan(-x) = _______ Csc(-x) = _______ Sec(-x) = _______ Sin(-x) = _______

Simplifying Trigonometric Expressions Simplify by factoring and using identities Sin 3 x + sinxcos 2 x

EXPLORATION In window [-2 π, 2 π ] by [-4, 4] graph y 1 = sinx and y 2 = NDER(sinx) 1. When the graph of y 1 = sinx is increasing, what is true about the graph of y 2 = NDER(sinx)? 2. When the graph of y 1 = sinx is decreasing, what is true about the graph of y 2 = NDER(sinx)? 3. When the graph of y 1 = sinx stops increasing and starts decreasing, what is true about the graph of y 2 = NDER(sinx)? 4. At the places where NDER(sinx) = ± 1, what appears to be the slope of the graph of y 1 = sinx? 5. Make a conjecture about what function the derivative of sine might be. Test your conjecture by graphing your function and NDER(sinx) in the same viewing window. 6. Now let y 1 = cosx and y 2 = NDER(cosx). Answer questions (1) through (5) without looking at the graph of NDER(cosx) until you are ready to test your conjecture about what function the derivative of cosine might be.