Chapter 5 Analytic Trigonometry Overview: 5.1 Using Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Solving Trig Equations 5.4 Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-sum Formulas
5.1 Using Fundamental Identities What You ll Learn: #103 - Recognize and write the fundamental trig identities. #104 - Use the fundamental trig identities to evaluate trig functions, simplify trig expressions, and rewrite trig expressions.
Fundamental Trig Identities (page 340) Quotient Identities tan θ = sin θ cos θ cot θ = cos θ sin θ Even Trig Functions cos θ = cos θ sec θ = sec θ Reciprocal Identities csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ Odd Trig Functions sin θ = sin θ csc θ = csc θ Pythagorean Identities tan θ = tan θ cot θ = cot θ sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ
Alternative Pythagorean Identities Pythagorean identities are sometimes used in radical form: sin x = ± 1 cos 2 x tan x = ± sec 2 x 1
Using Identities to Evaluate a Function Use the values sec u = 3 2 six trig functions. and tan u > 0 to find the values of all
Simplify a Trig Expression Simplify sin x cos 2 x sin x.
Verifying a Trig Identity Determine whether the equation appears to be an identity: cos 3x = 4 cos 3 x 3 cos x Check the table! y 1 = cos 3x y 2 = 4 cos 3 x 3 cos x Check the graphs!
Verifying a Trig Identity Establish the identity: sin θ + cosθ 1+cosθ sin θ = csc θ
Factoring Trig Expressions Factor each expression: A. sec 2 θ 1 B. 4 tan 2 θ + tan θ 3 C. csc 2 x cot x 3
Homework Page 345 #1,11,39,40,47,51,53,57
5.2 Verifying Trigonometric Identities What You ll Learn: #105 - Verify trigonometric identities.
Verifying a Trig Identity Establish the identity: sec2 θ 1 sec 2 θ = sin2 θ
Verifying a Trig Identity Establish the identity: 1 1 sin α + 1 1+sin α = 2 sec2 α
Verifying a Trig Identity Establish the identity: tan 2 x + 1 cos 2 x 1 = tan 2 x
Verifying a Trig Identity Establish the identity: tanθ + cotθ = secθcscθ
Verifying a Trig Identity Establish the identity: cot 2 θ 1+cscθ = 1 sinθ sinθ
Homework Page 353 #1-10, 41,42
5.3 Solving Trigonometric Equations What You ll Learn: #106 - Use standard algebraic techniques to solve trig equations. #107 - Solve trig equations of quadratic type. #108 - Solve trig equations involving multiple angles (solutions). #109 - Use inverse trig functions to solve trigonometric equations.
Trig Equations You are no longer EVALUATING trig functions, you are now SOLVING for solutions to the equations. In other words, you IGNORE all inverse restrictions!
Trig Equations sin θ = 1 2 What are all possible values of θ that would satisfy the above equation? θ = π 6, 5π 6, 13π 6 It could go on forever
Example Solve the equation: cos θ = 1 2 Two possible solutions: θ = π 3 and 5π 3 To display all possible solutions: θ = π 3 + k2π and θ = 5π 3 + k2π
Example Solve the equation: 2 sin θ + 3 = 0, 0 θ < 2π.
Example Solve the equation: sin 2θ = 1 2, 0 θ < 2π.
Example Solve the equation: tan θ π 2 = 1, 0 θ < 2π.
Example Solve the equation: sin 2θ π 2 = 1, 0 θ < 2π.
Example Solve the equation: csc 3θ 2 = 2, 0 θ < 2π.
Finding the other solutions sin θ = 1 2 π minus your answer θ = π 6 and π π 6 cos θ = 1 2 = 5π 6 make your answer negative, add 2π θ = π 3 and π 3 + 2π = 5π 6 tan θ = 1 θ = π 4 and π 4 + π = 5π 4 add π to your answer
Approximation Example Use a calculator to solve the equation: sin θ = 0. 3, 0 θ < 2π. Round any solution(s) to two decimal places.
Solve a Trig Quadratic Equation Solve the Equation on the interval 0 θ < 2π: 2 sin 2 θ 3 sin θ + 1 = 0 2x 2 3x + 1 = 0
Trig Equation Solve the equation on the interval 0 θ < 2π. 2 cos 2 θ + cos θ = 0
Trig Equation Solve the equation on the interval 0 θ < 2π. 3cos θ + 3 = 2 sin 2 θ
Trig Equation Solve the equation on the interval 0 θ < 2π. cot θ 1 cos θ + 1 = 0.
Trig Equation Solve the equation on the interval 0 θ < 2π. cos 2 θ + sin θ = 2
Trig Equation Solve the equation on the interval 0 θ < 2π. sec 2 x 2 tan x = 4
Approximating Trig Solutions Solve: 5 sin x + x = 3 Round solution(s) to two decimal places
Homework Page 364 #1, 7-10, 17-19, 31, 51
In-Class Review 5.1-5.3 Page 387 #1-12, 15, 16, 25-28, 37, 39, 42, 49, 52
5.4 Sum and Difference Formulas What You ll Learn: #110 - Use sum and difference formulas to evaluate trig functions, verify identities, and solve trig equations.
Exploration Does cos(x + 2) = cos x + cos 2? Graph each: y 1 = cos(x + 2) y 2 = cos x + cos 2
Sum/Difference Formulas for Cosines cos α + β = cos α cos β sin α sin β cos α β = cos α cos β + sin α sin β Examples: Find the exact solution of cos 75 o. cos 45 o + 30 o = cos45 o cos 30 o sin 45 o sin 30 o = 6 2 4 = 1 4 ( 6 2)
Example using cos(α β) Find the exact value of cos( π ). 12 cos ( 4π 12 3π 12 ) cos π 3 π 4 = cos π 3 cos π 4 + sin π 3 sin π 4 1 2 2 2 + 3 2 2 2 2 + 6 = 1 4 4 ( 6 + 2)
Sum/Difference Formulas for Sines sin α + β = sin α cos β + cos α sin β sin α β = sin α cos β cos α sin β 1. Find the exact value sin( 7π ). 12 2 + 6 = 1 4 4 ( 2 + 6) Examples: 2. Find the exact value of: sin 80 o cos 20 o cos 80 o sin 20 O. sin 80 o 20 o = sin 60 o = 3 2
Find Exact Values If it is known that sin α = 4 5, π find the exact value of: (a) (b) (c) cos α cos β cos(α + β) 2 < α < π, and that sin β = 2 5 5, π < β < 3π 2, (d) sin(α + β)
Establish an Identity Establish the identity: cos(α β) sin α sin β = cot α cot β + 1
Sum/Difference Formulas for Tangent tan α + β = tan α β = tan α + tan β 1 tan α tan β tan α tan β 1 + tan α tan β So Check this out! We can prove a trig property! tan(θ + π)
Example Find the exact value of the following trig function: tan( 5π 12 )
Find Exact Values If it is known that sin α = 3, 0 < α < π, and that sin β = 7, π < β < 3π, 5 2 25 2 find the exact value of: (a) tan(α + β) (b) tan(α β)
Homework Page 372 #3,7,10,20,23,35,40
Paper Clip Activity Copy this chart: Triangle # Side #1 Side #2 Side #3 Triangle? (Y or N) 1 2 3 4 5 6 7 8 Directions: Create 8 scalene triangles Side lengths must range from 1 to 6 No duplicates allowed
5.5 Multiple-Angle and Product-to- Sum Formulas What You ll Learn: #111 - Use multiple-angle formulas to rewrite and evaluate trig functions. #112 - Use power-reducing formulas to rewrite and evaluate trig functions. #113 - Use half-angle formulas to rewrite and evaluate trig functions. #114 - Use product-to-sum and sum-to-product formulas to rewrite and evaluate trig functions.
Double-Angle Formulas sin 2θ = 2 sin θ cos θ cos 2θ = cos 2 θ sin 2 θ cos 2θ = 1 2 sin 2 θ cos 2θ = 2 cos 2 θ 1
Examples Find the exact values using the double-angle formula if sin θ = 3 5, π 2 < θ < π A. sin(2θ) B. cos(2θ)
Double-Angle Formulas tan 2θ = 2 tan θ 1 tan 2 θ
Example Using Tangent Double-Angle Find the exact values using the double-angle formula if sin θ = 3 5 tan(2θ), π 2 < θ < π
Half-Angle Formulas sin( α 2 ) = ± cos( α 2 ) = ± tan( α 2 ) = ± 1 cos α 2 1 + cos α 2 1 cos α 1 + cos α
Alternative Half-Angle Tangent tan α 2 1 cos α = sin α = sin α 1 + cos α
Examples Using Half-Angle Find the exact value of: cos(15 o ) sin( 15 o ) tan(22.5 o )
Example If cos α = 3 5, π < α < 3π 2, find the exact value of: 1. sin α 2 2. cos α 2 3. tan α 2
Solving a Multiple-Angle Equation Solve 2 cos x + sin(2x) = 0.
Homework Page 382 #1-5,9,17,33-35,41,45,49
Chapter 5 Review Page 387 #12,20,38,41,43,51,63,67-72,77,85,89,99,103
Farkle Scoring Guide: Roll Points 1 100 5 50 Three 1 s 1,000 Three 2 s 200 Three 3 s 300 Three 4 s 400 Three 5 s 500 Three 6 s 600 Triples must be rolled on the same roll Pick up dice you didn't score to roll again If you roll and don t score anything, you lose the points from that round You must stop to bank the points before you lose them First player to 5,000 points wins Every player has to keep track of everyone s points