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# Analytic trigognometry

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Analytic Trigonometry

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### Transcript of "Analytic trigognometry"

1. 1. TOPIC 1.2 – ANALYTIC TRIGONOMETRY 1.2.1: The Inverse Sine, Cosine, and Tangent Functions 1.2.2: The Inverse Trigonometric Functions 1.2.3: Trigonometric Identities 1.2.4: Sum and Difference Formulas 1.2.5: Double-Angle and Half-Angle Formulas 1.2.6: Product-to-Sum and Sum-to-Product Formulas 1.2.7: Trigonometric Equations (I) 1.2.8: Trigonometric Equations (II) 1
2. 2. 2 Review of Properties of Functions and Their Inverses 1.2.1& 1.2.2: The Inverse Sine, Cosine, and Tangent Functions
3. 3. 3
4. 4. 1 sin xy sin 22 x y x x ysin sin1 The inverse sine function: The inverse sine function denoted by sine function from . Thus, is the inverse of the restricted 4
5. 5. x1 sin x1 sin Finding exact values of 1. Let = xsin2. Rewrite = asx1 sin xsin3. Use the exact values to find the value of that satisfies 2 3 sin 1 2 2 sin 1 Example: Find the exact value of; 1- 2- 5
6. 6. 6
7. 7. 1 cos xy cos x0 The inverse cosine function: The inverse cosine function denoted by restricted cosine function from . Thus, is the inverse of the y x x ycos cos1 0 y 1 1xwhere and Example: Find the exact value of; 1) 2 1 cos 1 7
8. 8. 8
9. 9. 1 tan xy tan 2 2 y The inverse tangent function: The inverse tangent function denoted by restricted tangent function from .Thus, is the inverse of the y x x ytan tan1 2 2 y xwhere and Example: Find the exact value of; 1) 1tan 1 9
10. 10. Composition of functions involving inverse trigonometric functions xx1 sinsin xxsinsin 1 2 , 2 Inverse properties 1.Sine function: for every x in the interval [-1,1] for every x in the interval xx1 coscos xxcoscos 1 ,0 2. Cosine function: for every x in the interval [-1,1] for every x in the interval xx1 tantan xxtantan 1 2 , 2 3. Tangent function: for every real number x for every x in the interval 10
11. 11. 7.0coscos 1 sinsin 1 2coscos 1 Example: 1-Find the exact value if possible; a) b) c) 4 3 tansin 1 2 1 sincos 1d) e) 2- If x > 0, write x1 tansec as an algebraic expression in x 11
12. 12. 1.2.3: Trigonometric Identities Fundamental trigonometric identities i ii iii.csc sin .sec cos .cot tan 1 1 1 Reciprocal Identities i ii.tan sin cos .cot cos sin Quotient Identities i ii iii .sin cos .tan sec . cot csc 2 2 2 2 2 2 1 1 1 Pythagorean Identities i ii iii iv v vi .sin( ) sin .cos( ) cos .tan( ) tan .csc( ) csc .sec( ) sec .cot( ) cot Even - Odd Identities 12
13. 13. Example: Verify the identity: Changing to sine and cosine 1) xxx sectancsc 2) xxxx cscsincotcos 3) xx xx xx cossin cscsec )csc(sec Using factoring 1) xxxx 32 sincossinsin 2) x x x x x csc2 sin cos1 cos1 sin Multiplying numerator and denominator by the same factor 1) x x x x cos sin1 sin1 cos Working with both sides separately 1) 2 tan22 sin1 1 sin1 1 13
14. 14. 1.2.4: Sum and Difference Formulas Sum and difference formulas for cosines and sines cos (A+ B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cos A sin B Example: 1.Using difference formula to find the exact value a) Give the exact value of  6090cos30cos using the sum and difference formula 12 5 sin 4612 5 b) Find the exact value of using the fact that 14
15. 15. tantan1 coscos cos 5 4 sin angle 2 1 sin angle 2. Verify the identity: 3. Suppose that for a quadrant II and for a quadrant I . Find the exact value of cos cos cos sin a) b) c) d) 15
16. 16. tan( ) tan tan tan tan1 tan( ) tan tan tan tan1 Sum and difference formulas for tangents 1- 2- Example: 1.Verify the identity: xx tantan 16
17. 17. 1.2.5: Double-Angle and Half-Angle Formulas cossin22sin 22 sincos2cos 2 tan1 tan2 2tan Double angle formulas: 1- 2- 3- 5 4 sin Example: 1- If and lies in quadrant II, find the exact value of; 2sin 2cos 2tana) b) c) 2. Find the exact value of 15sin15cos 22 17
18. 18. Using Pythagorean identity to write 2cos in terms of sine only: 22 sincos2cos 1cos22cos 2 2 sin212cos Three forms of the double angle formula for cos 1- 2- 3- Example: Verify the identity: 3 sin4sin33sin 2 2cos1 sin2 2 2cos1 cos2 2cos1 2cos1 tan2 Power reducing formulas x4 sin of trigonometric functions greater than 1 that does not contain powersExample: Write an equivalent expression for 18
19. 19. Half angle formulas cos1 cos1 2 tan; 2 cos1 2 cos; 2 cos1 2 sin The + or – in each formula is determined by the quadrant in which 2 lies Example: 1- Use cos 120º to find the exact value of cos 105º 2- Verify the identity: 2cos1 2sin tan 2 sin cos1 2 tan cos1 sin 2 tan Half angle formula for tan Example: Verify the identity: csccscsec sec 2 tan 19
20. 20. 1.2.6: Product-to-Sum and Sum-to-Product Formulas )]cos()[cos( 2 1 sinsin )]cos()[cos( 2 1 coscos )]sin()[sin( 2 1 cossin )]sin()[sin( 2 1 sincos 1- 2- 3- 4- Example: Express each of the following products as a sum or difference: a. sin 5x sin 2x b. cos 7x cos x 20
21. 21. 2 cos 2 sin2sinsin 2 cos 2 sin2sinsin 2 cos 2 cos2coscos 2 sin 2 sin2coscos Sum to Product Formulas: 1- 2- 3- 4- Example: 1- Express each sum or difference as a product a. sin 7x + sin 3x b. cos 3x +cos 2x 2- Verify the identity: x xx xx tan sin3sin cos3cos 21
22. 22. 1.2.7& 1.2.8: Trigonometric Equations • A trigonometric equation is an equation that contains a trigonometric expression. • To solve an equation containing a single trigonometric function:  Isolate the function on one side of the equation  Solve for the variable Finding all solutions of a trigonometric equation Example: Solve the equation: 3sin3sin5 xx Solving an equation with a multiple angle Example: Solve the equation: 32tan x 20 x 2 1 3 sin x 20 x 1- 2- 22
23. 23. 01sin3sin2 2 xx 20 x xxx sintansin 20 x Trigonometric equations quadratic in form Example: Solve the equation: Using factoring to separate 2 different trigonometric functions in an equation Example: Solve the equation: 0sin2cos xx 20 x 2 1 cossin xx 20 x 1sincos xx 20 x Using an identity to solve a trigonometric equation Example: Solve the equation: 1- 2- 3- 23
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