Chapter 5 – Analytic Trigonometry 5.1 – Fundamental Identities HW: Pg. 451 #1-8

Trigonometry Identities Reciprocal Identities csc θ= 1/sin θ sec θ=1/cos θ cot θ= 1/tan θ sin θ= 1/csc θ cos θ=1/sec θ tan θ=1/cot θ Quotient Identities Tan θ = sin θ /cos θ cot θ=cos θ/sin θ

Pythagorean Identities Try to come up with an identity using the pythagorean theorem (a2 + b2 = c2) and the unit circle

Pythagorean Identities Cos2θ + Sin2θ = 1 1 + tan2θ = Sec2θ Cot2θ + 1 = csc2θ

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

Cofunction Identities ANGLE A: ANGLE B: sinA = y/r sinB = CosA = x/r CosB = tanA = y/x tanB = cscA = r/y cscB = secA = r/x secB = cotA = x/y cotB =

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) = _______ cot(-x) = _______

Using Identities If cos θ = 0.34, find sin(θ - π/2).

Simplifying Trigonometric Expressions Simplify by factoring and using identities Sin3x + sinxcos2x

Simplify by expanding and using identities [(secx + 1)(secx - 1)]/sin2x

Simplify by combining fractions and using identities

Solving Trigonometric Equations Find all values of x in the interval [0,2π) that solve cos3x / sinx = cotx

Solving a Trig Equation by factoring Find all solutions to the trigonometric equation 2sin2x + sinx = 1

PRACTICE 5.1 – Pg. 451-452 #9-16, 23-38 (all) Use Identities to simplify

5.2 – Proving Trigonometric Identities HW: PG. 460 #12-34e

Proving an Algebraic Identity Prove the algebraic identity : (x2-1)/(x-1) – (x2-1)/(x+1) = 2

General Proof Strategies I The proof begins with the expression on one side of the identity. The proof ends with the expression on the other side The proof in between consists of showing a sequence of expressions, each one easily seen to be equivalent to its previous expression.

Proving an Identity: tan x + cot x = sec x csc x

General Strategies II Begin with the more complicated expression and work toward the less complicated expression. If no other move suggests itself, convert the entire expression to one involving sines and cosines. Combine fractions by combining them over a common denominator.

Identifying and Proving an Identity Match the function f(x)=1/(secx - 1) + 1/(secx + 1) with (i) 2cot x csc x or (ii) 1/secx

Setting up a Difference of Squares Prove the identity: cost/(1 – sint) = (1 + sint)/cost

General Strategies III Use the algebraic identity (a + b)(a – b) = a2 – b2 to set up applications of the Pythagorean identities. Always keep in mind what your goal is (expression), and favor manipulations that bring you closer to your goal.

Working from Both Sides Prove the identity: Cot2u/(1 + cscu) = (cotu)(secu – tanu)

Disproving Non-Identities - EXPLORATION Prove or disprove that this is an identity: cos2x = 2cosx Graph y=cos2x and y=2cosx in the same window. Interpret the graphs to make a conclusion about whether or not the equation is an identity. With the help of the graph, find a value of x for which cos2x ≠ 2cosx. Does the existence of the x value in part (2) prove that the equation is not an identity? Graph y = cos2x and y = cos2x – sin2x in the same window. Interpret the graphs to make a conclusion about whether or not cos2x = cos2x – sin2x is an identity. Do the graphs in part (4) prove that cos2x = cos2x – sin2x is an identity? Explain your answer.