NOTES 10: ANALYTIC TRIGONOMETRY

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1 NOTES 0: ANALYTIC TRIGONOMETRY Name: Date: Period: Mrs. Nguyen s Initial: LESSON 0. USING FUNDAMENTAL TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sin csc cos sec tan cot csc sin sec cos cot tan Quotient Identities tan sin cos cot cos sin Pythagorean Identities Cofunction Identities Even/Odd Identities sin cos sin 90 cos sin cos tan 90 cot tan cot sec 90 csc sec csc Even Functions cos( t) cost sec( t) sect tan sec cot csc cos 90 sin cos sin cot 90 tan cot tan csc 90 sec csc sec Odd Functions sin( t) sint csc( t) csct tan( t) tant cot( t) cott Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page

2 . Practice Problem : 3 Given sin x and cos x. Find:. Practice Problem : 3 Given cos x 5 4 and cos x. Find: 5 tan x cot x sin x tan x sin x csc x sec x csc x sec x cot x Practice Problems: Simplify the following trig expressions 3. sin tan cos 4. tan x 5. cos x sin x sinx cosx 6. sin x cos x Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page

3 Practice Problems: Factor each expression. 7. 4tan tan sin 8 Practice Problem 9: Rewrite sinx so that it is not in fractional form. Practice Problem 0: Use the trig substitution x 3cos, 0 to express 9 x as a trig function of. Practice Problem : Use the trig substitution to write the algebraic equation as a trig function of where. Then find sin and cos. 64, cos x x Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page 3

4 LESSON 0. 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 n 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 csc sec cot. sin cos sec cos sin Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page 4

5 sin sin cos cos 4. tan x cot x sec xcsc x 5. cot sin csc sin 6. cos y sec ytan y siny 7. cos cos cos sin 8. sec x cot x Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page 5

6 LESSON 0.3 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. Nguyen Honors PreCalculus Chapter 0 Notes Page 6

7 Practice Problems: Find all solutions of the equation in the interval 0, cos x sin xtan x sec x sec x 8. sin x 3sin x tan x 6tan x tan3x. 3tan x 0 Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page 7

8 LESSON 0.4 SUM AND DIFFERENCE FORMULAS Sum and Difference Formulas sin u v sinucosvcosusin v sin u v sinucosv cosusin v cos uv cosucosvsinusin v cos uv cosucosvsinusin v tan tan tanu tan v uv tanutanv tanu tan v uv tanutanv Practice Problem : Find the exact value. a. sin05 b. cos05 c. tan05 Practice Problem : Find the exact value. a. sin b. cos c. tan Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page 8

9 Practice Problem 3: Write the expression as the sine, cosine, or tangent of an angle. a. cos5cos5sin 5sin5 b. tan x tan x tan x tan x Practice Problem 4: Find the exact value of the expression. a. cos5cos60sin5sin 60 b. tan 5tan0 tan5tan0 Practice Problems: Verify the identity 5. tan tan 4 tan 6. sin3 x sin x Mrs. Nguyen Honors PreCalculus Chapter 0 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. secv u b. cotu v Practice Problems: Find all solutions of the equation in the interval 0,. 8. sinx sinx 3 3 tan cosx 0 9. x Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page 0

11 LESSON 0.5 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. Nguyen Honors PreCalculus Chapter 0 Notes Page

12 Practice Problem 4: Use the following to find sin, cos, and tan. Given: 3 cot 4;. a. sin b. cos c. tan 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. Nguyen Honors PreCalculus Chapter 0 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 b. csc 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 sin cos tan Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page 3

14 Sum-to- Product Formulas x y x y sin xsin y sin cos x y x y sin xsin y cos sin x y x y cos xcos y cos cos x y x y cos xcos y sin sin Practice Problem 0: Write sums as products sin sin Practice Problem : Find all solutions in the interval 0,. 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 sec sec Mrs. Nguyen Honors PreCalculus Chapter 0 Notes Page 4