FUNDAMENTAL TRIGONOMETRIC INDENTITIES 1 = cos. sec θ 1 = sec. = cosθ. Odd Functions sin( t) = sint. csc( t) = csct tan( t) = tant

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1 NOTES 8: ANALYTIC TRIGONOMETRY Name: Date: Period: Mrs. Nguyen s Initial: LESSON 8.1 TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sinθ 1 cscθ cosθ 1 secθ tanθ 1 cotθ cscθ 1 sinθ secθ 1 cosθ cotθ 1 tanθ Quotient Identities tanθ sinθ cosθ cotθ cosθ sinθ Pythagorean Identities Cofunction sin θ + cos θ 1 sin( 90 θ ) Identities cosθ π sin θ cos θ tan 90 θ cotθ π tan θ cot θ sec 90 θ cscθ π sec θ csc θ 1+ tan θ sec θ 1+ cot θ csc θ ( θ ) cos 90 sinθ π cos θ sin θ cot 90 θ tanθ π cot θ tan θ csc 90 θ secθ π csc θ sec θ Even/Odd Identities Even Functions cos( t) cost sec( t) sect Odd Functions sin( t) sint csc( t) csct tan( t) tant cot( t) cott When simplifying trigonometry epressions try to get the epression down to a single trig function. Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 1

2 Practice Problems: Simplify each trig epression. 1. sin sec cot. cos csc sec 3. sec cos cos 4. sin cos + csc sec Review Conditional Equation Only true for some of the value in its domain. sin 0 nπ where n is an integer Identity True for all real values in its domain. sin 1 cos True for all real numbers. Guidelines for Verifying Trig Identities 1. 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 epression, 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 epression 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. NOTE: Never move terms from one side of the equation to the other side. Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page

3 Practice Problems: Verify the identity cot + tan csc sec (1 cos )(1 cos ) sin 7. sin 1 cos 1+ cos sin 8. cscθ cos θ + sinθ cscθ 9. cot (tan sin + cos ) csc sin cos cos 1 sin Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 3

4 sec cos sec + 1+ cos sec cos 1. sin (csc sin ) cos 13. sin + cos + tan sec 14. sec + csc sin + cos tan + cot 15. tan sin tan sin tan + sin tan sin 16. cot tan cot 1 tan Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 4

5 LESSON 8. ADDITION AND SUBTRACTION 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 + 1 tanutanv ( u v) tanu tan v 1 + tanutanv ( u v) Practice Problem 1: Find the eact value. a. sin105 b. cos105 c. tan105 Practice Problems: Use the sum and difference of angles formulas to find the eact value of each epression.. sin cos tan cos 5 π 1 6. sin 7 π 1 7. tan 11 π 1 Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 5

6 8. cos10 cos80 sin10 sin π π tan + tan 18 9 π π 1 tan tan 18 9 Practice Problems: Prove the given identity. 10. cos( θ + 90 ) sinθ π 11. sin + cos 1. sec( θ + 90 ) cscθ 13. csc( θ 90 ) secθ 14. sin( y) tan tan y 15. cos cos y cot cot y + 1 cot( y) cot y cot Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 6

7 Sums of Sines and Cosines (proof on page 586 in the tetbook) When given an epression of the form Asin + Bcos, you can transform it into the form ksin( + φ) by letting k A + B, A B cosφ and sinφ. k k To find the angle φ, look at the values of sine and cosine and determine what quadrant the angle lies in. Then find the angle measure in that quadrant. Practice Problems: Write the epression in terms of sine only sin cos 17. 3cos sin 18. 6cos ( sin) 19. sin+ cos 0. 3sinπ + 3 3cosπ Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 7

8 LESSON 8.3 DOUBLE-ANGLE, HALF-ANGLE, AND PRODUCT-SUM FORMULAS Double Angle Formulas sin u sinucosu u u u u u cos cos sin cos 1 1 sin tan u tanu 1 tan u Practice Problem 1: Use the following to find sin θ, cos θ, and tan θ. Given: 1 sin, is in quadrant I. 13 a. sin θ b. cosθ c. tan θ Practice Problem : Use the following to find sin θ, cos θ, and tan θ. Given: 1 cos, tan < a. sin θ b. cosθ c. tan θ Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 8

9 Practice Problem 3: Use the following to find sin θ, cos θ, and tan θ. Given: 1 tan A, sin < 0. 5 a. sin θ b. cosθ c. tan θ Practice Problems: Simplify each epression. 4. sin35 cos35 5. cos 5 1 θ 6. tan50 1 tan 50 Practice Problems: Prove the following identities. 7. csc θ cot θ tanθ tan sec 1 tan Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 9

10 9. sin cot cos cos cos sin sin + cos Half-Angle Formulas u 1 cosu sin ± u 1+ cosu cos ± u tan 1 cosu sin u sinu 1 + cosu Practice Problems: Calculate sin,cos,and tan for the angle described cos ;0 < < sin cos tan 1. 3 sin θ ;70 < θ < sin cos tan 13. cos θ ;180 < θ < 70 3 sin cos tan Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 10

11 Practice Problems: Find the eact value of each epression by using the half-angle formulas. 14. sin π cos sin π cos tan π tan 8 Practice Problems: Prove the following identities sin cos tan tan sin tan + tan. 1 + cos sin 1 cos + csc 1 cos sin Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 11

12 LESSON 8.4 INVERSE TRIGONOMETRIC FUNCTIONS Graph of Sine Function: y sin Graph of Inverse Sine Function: y arcsin sin y π π Domain:, Range: [ 1, 1] y π π 1 π π On the interval,, y sin is increasing and has a unique inverse function called Inverse Sine Function. Domain: Range: Graph of Cosine Function: y cos Graph of Inverse Cosine Function: y arccos cos Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 1

13 Domain: [ ] 0, π Range: [ 1, 1] y 0 1 π 0 π -1 Domain: Range: y 0, π, y cos is decreasing and has a unique inverse function called Inverse Cosine Function. On the interval [ ] Graph of Tangent Function: y tan Graph of Inverse Tangent Function: y arctan tan π π Domain:, Range: (, ) y π π 1 4 Domain: Range: y π π On the interval,, y tan is increasing and has a unique inverse function called Inverse Tangent Function. Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 13

14 Definitions of the Inverse Trig Functions Function Domain Range y arcsin iff sin y 1 π π y y arccos iff cos y 1 0 y π y arctan iff tan y < < π π < y < Remember sin( angle) cos( angle) tan( angle) ratio ratio ratio arcsin arccos arctan sin cos tan ratio ratio angle ratio ratio angle ratio ratio angle Practice Problem 1: Find the eact value. a. 1 arcsin b. sin 1 3 c. sin d. arccos e. arccos( 1) f. tan ( 0) g. arctan( 1) h. arccos i. arctan( 3) Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 14

15 Practice Problem : Evaluate in radians. Round to 3 decimal places. a. arcsin( 0.45 ) b. sin (.314) c. arctan( π ) d. 1 arccos 3 e. cos ( π ) f. 95 tan 7 Review Inverse functions have the properties: Inverse Properties of Trig Functions ( 1 ) f f If 1 and ( ) and sin( arcsin ) f f π π y, then arcsin sin y and y Eample: If 1 and 0 y π, then cos arccos arccos cos y and y Eample: If is a real number and tan( arctan ) π π y, then and arctan( tan y) y Eample: Practice Problem 3: Find the eact value a. tan arctan( 5) b. 5π arcsin sin 3 c. cos cos ( π ) Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 15

16 Practice Problems: Find the eact value tan arccos 3 3 cos arcsin 5 5 cot arctan 8 Practice Problem 7: Find an algebraic epression that is equivalent to the epression. a. sin arctan( ) b. sec arctan 3 Practice Problem 8: Find the eact value of each epression, if it is defined. a. sin sin cos b. cos( tan ) c. ( 1 cos cos sin 1 + ) d. tan ( sin ) Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 16

17 LESSON 8.5 TRIGONOMETRIC EQUATIONS Tips for Solving Trig Equations 1. Get an equation (or equations) in which one function equals a constant.. Substitute in any obvious trig identities. 3. Reduce the number of different functions (get the equation so it only has 1 trig function). 4. Do any obvious algebra (get terms into 1 fraction, factor, use the quadratic formula). Remember, in order to factor, the equation MUST be set equal to 0!! 5. Do not divide both members by a variable (you can NOT divide by a trig function) NOTE: Always find the general solution first!! This is especially important if the angle () has a coefficient (multiple angle)! Practice Problems: Solve the equation in the interval [0, π ). 1. cos sec csc sin 1 5. (3 tan )(tan 3) 0 6. tan + tan 0 7. cos cos Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 17

18 9. sin sin cos sin cos sin 3 cos + cos3sin 1 1. sin sin 13. cos sin 14. cos + 5cos cot sin cot 16. 3cos 8cos tan 8tan sin cos Mrs. Nguyen Honors Algebra II Chapter 8 Notes Page 18