6.1 Reciprocal, Quotient, and Pythagorean Identities

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1 Chapte 6 Tigonometic Identities Recipocal, Quotient, and Pthagoean Identities Wam-up Wite each epession with a common denominato. Detemine the estictions. a c a a) b d b) b c d c) a 1 c b c b a Definition Tigonometic identit The equation tan cos sin is identit because it is tue fo all values of ecept k, whee k is an intege (whee tan is not defined). You can pove this is tue b gaphing 1 tan cos and sin [ ] [ ] [ ] [ ] Pe-Calculus 1

2 Chapte 6 Tigonometic Identities The most complete method fo poving tigonometic identities uses algeba. This method can involve simplifing, factoing, and e-witing epessions. In Chapte 4, si tigonometic functions wee defined in tems of angle in standad position in a cicle with a adius,, and a teminal point P(, ) on the cicle. P(, ) sin cos tan csc sec cot Some basic tigonometic identities (on fomula sheet) Recipocal Identities 1 csc sec sin 1 cos 1 cot cot Quotient Identities sin tan cos cos cot sin Pthagoean Identities θ sin sin cos cos Pthagoean Theoem tells us that: o o Note: Pthagoean Theoem is the onl one of the identities that ou can manipulate. Pe-Calculus 1

3 Identifing non-pemissible values then Poving an Identit A tigonometic epession, like an algebaic epession, cannot have a zeo in the denominato. Chapte 6 Tigonometic Identities 3 Eample 1: a) Detemine the non-pemissible values in degees fo the equation tan cos sin. b) Numeicall veif that 45 is a solution of the equation. c) Pove algebaicall that tan cos sin Pe-Calculus 1

4 4 Chapte 6 Tigonometic Identities sin tan Eample : Fo the epession: tan, cos 1 a) Identif the estiction on the vaiable. b) Pove the identit. sin tan tan cos 1 Pe-Calculus 1

5 Chapte 6 Tigonometic Identities 5 Simplifing Epessions Eample 3: Simplif the epession cos 1 cos Seveal stategies that ae often successful wee used in this poof. Descibe the stategies used. Thee is often moe than one coect wa to pove an identit. Sometimes it helps to wok on both sides of the equation until the simplif to the same epession. Based on epeience the moe complicated-looking side is the best place to stat. Eample 4: Pove csc 1 cot. State an estictions on θ. cot csc 1 Note: In this eample the two sides appea smmetical: thee is no hade side to stat on! Epessions such as ( sin 1) and ( sin 1) ae called the conjugates of each othe. Multipling them sometimes poduces a Pthagoean Identit: ( sin 1)( sin 1) sin sin sin 1 sin 1 cos Use this idea as a hint. Pe-Calculus 1

6 6 Chapte 6 Tigonometic Identities csc 1 cot cot csc 1 Tips The use of the conjugate in some poofs (see Eample 4) is based on the pinciple of multipling b 1. Do this to one side of the poof onl. Once an identit is established, it can be eaanged. Fo eample cot csc 1 is just anothe vesion of cot 1 csc, and an eaangement can be used in futue poofs. Do not combine moe than one step in a poof on the same line. You easoning will not be clea and ou ma be penalized. If second degee tems ae involved (e. sin ), conside using the Pthagoean Identities o factoing. Avoid using squae oots. Recipocal and Quotient Identities can be genealized; fo eample: 1 5 3cos 3 csc, 5csc, 3cot o 3cot sin sin sin tan Avoid common mistakes, such as cos cos, sin cos 1, sin sin sin. Pe-Calculus 1

7 Chapte 6 Tigonometic Identities 7 You T sec sec 1 Pove. State an estictions on θ. 1 cos sin Assignment: woksheet (do odd numbes) Pe-Calculus 1

8 8 Chapte 6 Tigonometic Identities 6. Sum, Diffeence and Double-Angle Identities Sum and Diffeence Identities sin( ) sin cos cos sin sin( ) sin cos cos sin cos( ) cos cos sin sin cos( ) cos cos sin sin tan tan tan( ) 1 tan tan tan tan tan( ) 1 tan tan Eample 1: Epess the following as a tigonometic function of a single angle: sin cos cos sin. 5 5 Eample : Conside the identit sin( 90 ) sin. Pove the identit algebaicall. Pe-Calculus 1

9 Chapte 6 Tigonometic Identities 9 Eample 3: Find the eact value of 7 cos without a calculato. 1. Eample 4: If sin A and 3 Oh no! That means ou will need to calculate the tig atios fo the following angles without a calculato 5 7 ±15 o, ±75 o, ±105 o Fo angles, ±165 o, ±195 o,., 1 1 ou ma have to add o subtact moe than once. 3 cos B. Both A and B ae in Quadant, evaluate cos( A B). 5 Assignment: woksheet (do 1-39 odd poblems and 40-4) Pe-Calculus 1

10 10 Chapte 6 Tigonometic Identities 6. Sum, Diffeence and Double-Angle Identities (continue ) Double Angle Identities sin sin cos cos cos sin cos 1 1 sin tan tan 1 tan Eample 1: If 1 sin A and A is in Quadant 3, evaluate tan A. 3 Eample : Wite 30 sin Acos A as a single tigonometic atio solution. You t Wite each of the following as a single tigonometic atio solution. a) cos 5 sin 5 b) 3 6sin 4 c) sin cos Pe-Calculus 1

11 Chapte 6 Tigonometic Identities 11 Eample 3: If sin A and A is in Quadant, evaluate 5 a) cos A b) sin A Pe-Calculus 1

12 1 Chapte 6 Tigonometic Identities Eample 4: Pove the identit, 1 cos sin tan. 1 cos sin tan Assignment: woksheet Pe-Calculus 1

13 Chapte 6 Tigonometic Identities Poving Identities Eample 1: Pove the identit tan 1 sec 1 cos cos tan 1 sec 1 cos cos Pe-Calculus 1

14 14 Chapte 6 Tigonometic Identities Eample : Pove the identit 1 sec sin sec 1 sin cos 1 sec sin sec 1 sin cos Pe-Calculus 1

15 Chapte 6 Tigonometic Identities 15 Eample#: Pove the identit sin cos 4sin 1 sin cos cos sin 1 sin cos 4sin 1 sin cos cos sin 1 Pe-Calculus 1

16 16 Chapte 6 Tigonometic Identities Recall: On page 5 of the notes, stategies when woking with poofs. Assignment: page 314 #1-4, 6, 7, 8, 10, 11 Pe-Calculus 1