While you wait: For a-d: use a calculator to evaluate: Fill in the blank.
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1 While you wait: For a-d: use a calculator to evaluate: a) sin 50 o, cos 40 o b) sin 25 o, cos65 o c) cos o, sin 79 o d) sin 83 o, cos 7 o Fill in the blank. a) sin30 = cos b) cos57 = sin
2 Trigonometric Identities and Equations Section 8.4
3 Cofuntion Relationships
4 UC revisited y (,y) y Pythagorean Theorem: 2 + y 2 =
5 UC revisited y (cos,sin ) sin θ cos Pythagorean Theorem:cos 2 θ + sin 2 θ =
6 The trig relationships: cos 2 θ + sin 2 θ = cos 2 θ = sin 2 θ sin 2 θ = cos 2 θ
7 An identity is an equation that is true for all values of the variables. Difference between identity and equation: An identity is true for any value of the variable, but an equation is not. For eample the equation 3=2 is true only when =4, so it is an equation, but not an identity.
8 What are identities used for? They are used in simplifying or rearranging algebraic epressions. By definition, the two sides of an identity are interchangeable, so we can replace one with the other at any time. In this section we will study identities with trig functions.
9 The trigonometry identities There are dozens of identities in the field of trigonometry. Many websites list the trig identities. Many websites will also eplain why identities are true. i.e. prove the identities. For an eample of such a site: click here
10 Trigonometric Identities Quotient Identities tan sin cos cot cos sin Reciprocal Identities sin csc Pythagorean Identities cos sec tan cot sin 2 + cos 2 = tan 2 + = sec 2 cot 2 + = csc 2 sin 2 = - cos 2 cos 2 = - sin 2 tan 2 = sec 2 - cot 2 = csc
11 Where did our pythagorean identities come from?? Do you remember the Unit Circle? What is the equation for the unit circle? 2 + y 2 = What does =? What does y =? (in terms of trig functions) sin 2 θ + cos 2 θ = Pythagorean Identity!
12 Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos 2 θ sin 2 θ + cos 2 θ =. cos 2 θ cos 2 θ cos 2 θ tan 2 θ + = sec 2 θ Quotient Identity another Pythagorean Identity Reciprocal Identity
13 Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin 2 θ sin 2 θ + cos 2 θ =. sin 2 θ sin 2 θ sin 2 θ + cot 2 θ = csc 2 θ Quotient Identity a third Pythagorean Identity Reciprocal Identity
14 Simplifying Trigonometric Epressions Identities can be used to simplify trigonometric epressions. Simplify. a) cos sin tan cos sin sin cos cos sin2 cos cos 2 sin 2 cos cos b) cot 2 sin 2 cos 2 sin 2 cos 2 cos 2 sin 2 cos 2 sin 2 sec csc
15 Practice Problems for Day : refer to class handout.
16 While you wait Factor: a) 2 4 b) 2 36 c) 2 d) 2 Identify as True or False: A. cos θ = cos (θ) B. sin θ = sin(θ) C. tan θ = tan(θ)
17
18 Proving a Trigonometric Identity:. Transform the right side of the identity into the left side, 2. Vice versa (Left side to Right ) We do not want to use properties from algebra that involve both sides of the identity.
19 Guidelines for Proving Identities:. It is usually best to work on the more complicated side first. 2. Look for trigonometric substitutions involving the basic identities that may help simplify things. 3. Look for algebraic operations, such as adding fractions, the distributive property, or factoring, that may simplify the side you are working with or that will at least lead to an epression that will be easier to simplify.
20 4. If you cannot think of anything else to do, change everything to sines and cosines and see if that helps. 5. Always keep an eye on the side you are not working with to be sure you are working toward it. There is a certain sense of direction that accompanies a successful proof. 6. Practice, practice, practice!
21 Prove cota( + tan 2 A) tana = csc 2 A cota(sec 2 A) tana = csc 2 A Pythagorean Relationship
22 cosa sina ( cos 2 A ) sina cosa =csc 2 A Definition of trig Functions sinacosa sina cosa = csc 2 A Reduce
23 cosa sin 2 AcosA =csc2 A Reduce sin 2 A = csc2 A Def of trig function. csc 2 A = csc 2 A
24 Practice Problems Day 2 Sec 8- Written Eercises page 32 #3-9 odds; odds Eit Question: #3b the handout. A complete, step by step solution must be included.
25 Using the identities you now know, find the trig value..) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ. sec cos sin 2 cos 2 sin sin sin sin 4 5 csc sin
26 3.) sinθ = -/3, find tanθ tan 2 sec 2 tan 2 (3) 2 tan 2 8 tan ) secθ = -7/5, find sinθ tan 2 8
27 Simplifing Trigonometric Epressions c) ( + tan ) 2-2 sin sec ( tan) 2 2 sin cos 2 tan tan 2 2 sin cos tan 2 2tan 2 tan sec 2 d) csc tan cot sin sin cos cos sin sin sin 2 cos 2 sincos sin sin cos sin cos sin cos
28 Simplify each epression. sin cos sin sin sin cos cos sec cos sin sin cos cos cos sin sin cos 2 sin sin2 sin cos 2 sin 2 sin sin csc
29 Simplifying trig Identity Eample: simplify tancos sin tan cos cos tancos = sin
30 Simplifying trig Identity Eample2: simplify sec csc cos sec csc cos sin sin = cos = = tan sin
31 Simplifying trig Identity Eample2: simplify cos 2 - sin 2 cos cos 2 - sin 2 cos = sec
32 Eample Simplify: = cot (csc 2 - ) Factor out cot = cot (cot 2 ) Use pythagorean identi = cot 3 Simplify
33 Simplify: Eample = sin (sin ) + cos cos = sin 2 + (cos ) cos cos cos = sin 2 + cos 2 cos = cos = sec Use quotient identity Simplify fraction with LCD Simplify numerator Use pythagorean iden Use reciprocal identity
34 Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity
35 Practice
36 One way to use identities is to simplify epressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let s see an eample of this: tan sin cos substitute using each identity csc sin simplify sin cos sin cos Simplify: tan csc sec cos cos sec cos
37 Another way to use identities is to write one function in terms of another function. Let s see an eample of this: Write the following epression in terms of only one trig function: 2 cos sin 2 = sin sin 2 = sin sin 2 This epression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute. 2 2 sin cos cos sin 2 2
38 (E) Eamples Prove tan() cos() = sin() LS tan cos LS sin cos cos LS sin LS RS 38
39 (E) Eamples Prove tan 2 () = sin 2 () cos -2 () RS RS RS RS RS RS RS sin 2 sin cos 2 sin 2 cos 2 sin sin cos tan 2 LS cos cos
40 (E) Eamples Prove tan tan sin cos LS LS LS LS LS LS LS tan tan sin cos sin cos sin cos cos sin sin sin cos cos cos sin 2 2 sin cos cos sin cos sin RS 40
41 (E) Eamples Prove 2 sin cos cos LS LS LS LS 2 sin cos 2 cos cos ( cos )( cos ) ( cos ) cos LS RS 4
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