( ) ( ) ( ) FINAL REVIEW: ANALYTIC TRIGONOMETRY REVIEW. θ = REVIEW LESSON 1 USING FUNDAMENTAL TRIGONOMETRIC IDENTITIES

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1 FINAL REVIEW: ANALYTIC TRIGONOMETRY REVIEW Name: Date: Period: REVIEW LESSON USING FUNDAMENTAL TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities cscθ cosθ secθ tanθ cotθ cscθ secθ cosθ cotθ tanθ Quotient Identities tanθ cosθ cotθ cosθ Pythagorean Identities Cofunction Identities Even/Odd Identities sin θ + cos θ ( θ ) sin 90 cosθ π sin θ cos θ tan 90 θ cotθ π tan θ cot θ sec 90 θ cscθ π sec θ csc θ Even Functions cos( t) cost sec( t) sect + tan θ sec θ + cot θ csc θ ( θ ) cos 90 π cos θ sin θ cot 90 θ tanθ π cot θ tan θ csc 90 θ secθ π csc θ sec θ Odd Functions sin( t) sint csc( t) csct tan( t) tant cot( t) cott Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page

2 . Practice Problem : 3 Given sin x and cos x. Find:. Practice Problem : π 3 Given cos x 5 4 and cos x. Find: 5 tan x csc x cot x sec x sin x csc x tan x sec x π cot x sin x Practice Problems: Simplify the following trig expressions 3. sin β tan β + cos β 4. tan x + 5. cos x + sin x + + sinx cosx 6. ( sin x cos x) + Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page

3 Practice Problems: Factor each expression. 7. 4tan θ tanθ sin θ 8 Practice Problem 9: Rewrite + sinx so that it is not in fractional form. π Practice Problem 0: Use the trig substitution x 3cos θ, 0 < θ < to express 9 x as a trig function of θ. Practice Problem : Use the trig substitution to write the algebraic equation as a π π trig function of θ where < θ <. Then find and cosθ. 6 4, cos x x θ Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 3

4 REVIEW LESSON VERIFYING TRIGONOMETRIC IDENTITIES Review Conditional Equation Only true for some of the value in its domain. Identity True for all real values in its domain. sin x 0 x nπ sin x cos x where n is an integer True for all real numbers x. Guidelines for Verifying Trig Identities. Work with one side of the equation at a time. It is often better to work with the more complicated side first.. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even paths that lead to dead ends give you insights. Practice Problems: Verify the identities. csc θ cscθ secθ cotθ. + cosθ + secθ cosθ + Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 4

5 sin α sin α cos α cos α 4. tan x + cot x sec xcsc x 5. cot θ + cscθ 6. cos y sec y+ tan y siny 7. cosθ cosθ + cosθ 8. π sec x cot x Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 5

6 REVIEW LESSON 3 SOLVING TRIGONOMETRIC EQUATIONS Practice Problems: Solve the equations.. cosx + 0. sin x + sin x 3. 3sec x cot cos cot x x x 5. Solve the equation by squaring and converting to quadratic type. Check your solutions. cos x + sin x Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 6

7 Practice Problems: Find all solutions of the equation in the interval [ 0, π ) cos x + sin xtan x sec x sec x 8. sin x 3sin x tan x 6tan x tan( 3x ). 3tan x 0 Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 7

8 REVIEW LESSON 4 SUM AND DIFFERENCE FORMULAS Sum and Difference Formulas sin u+ v sinucosv+ cosusin v sin u v sinucosv cosusin v cos u+ v cosucosv sinusin v cos u v cosucosv+ sinusin v tan tan tanu+ tan v + tanutanv ( u v) tanu tan v + tanutanv ( u v) Practice Problem : Find the exact value. a. sin05 b. cos05 c. tan05 Practice Problem : Find the exact value. a. sin π b. cos π c. tan π Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 8

9 Practice Problem 3: Write the expression as the sine, cosine, or tangent of an angle. a. cos5 cos5 sin 5 sin5 b. tan x + tan x tan x tan x Practice Problem 4: Find the exact value of the expression. a. cos5 cos60 + sin5 sin 60 b. tan 5 + tan0 tan5 tan0 Practice Problems: Verify the identity 5. π tanθ tan θ 4 + tan θ 6. sin( 3 x) π sin x Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 9

10 Practice Problem 7: Find the exact value of the trig function given that 3 and cosv. (Both u and v are in Quad II) 5 5 sinu 3 a. sec( v u) b. cot( u+ v) Practice Problems: Find all solutions of the equation in the interval [ 0, π ). π π 8. sin x+ + sin x 3 3 π tan + cos x ( x π ) Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 0

11 REVIEW LESSON 5 MULTIPLE-ANGLE AND PRODUCT-TO-SUM FORMULAS Double Angle Formulas sin u sinucosu cos cos sin cos sin u u u u u tan u tanu tan u Practice Problems: Find all solutions. sin 4x sin x. sin x+ cosx Practice Problem : Use a double-angle formula to rewrite the expression: cos x + sin x cos x sin x Practice Problem 3: Express sin3x in terms of sin x. Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page

12 Practice Problem 4: Use the following to find sin θ, cos θ, and tan θ. Given: 3π cotθ 4; < θ < π. a. sin θ b. cosθ c. tan θ Power- Reducing Formulas cosu sin u + cosu cos u cosu tan u + cosu Practice Problems: Rewrite the expression in terms of the first power of the cosine. 5. sin xcos x 6. 4 sin x Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page

13 Half-Angle Formulas u cosu sin ± u + cosu cos ± u tan cosu sin u sinu + cosu Practice Problem 7: Find the exact value of the trig functions a. cos θ b. csc θ θ 5 8 Practice Problems: Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle ' sin cos tan 9. 3π 8 sin cos tan Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 3

14 Sum-to- Product Formulas x + y x y sin x+ sin y sin cos x + y x y sin x sin y cos sin x + y x y cos x+ cos y cos cos x + y x y cos x cos y sin sin Practice Problem 0: Write sums as products sin( α + β) sin( α β) Practice Problem : Find all solutions in the interval [ 0, π ). sin 3x sin x 0 Practice Problem : Find all solutions. sin5x + sin3x 0 Practice Problems: Verify the identities 3. sint+ sin3t cost+ cos3t tan t 4. sec θ secθ sec θ Mrs. Atkinson Math Analysis Chapter 5 Review Notes Page 4