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While you wait: For a-d: use a calculator to evaluate: a) sin 50π , cos 40π b) sin 25π , πππ 65π c) πππ 11π , sin 79π d) sin 83π , cos 7π Fill in the blank. a) π ππ30° = cos ___° b) πππ 57° = sin ___° Trigonometric Identities and Equations Section 8.4 Cofuntion Relationships UC revisited y (x,y) 1 y x x Pythagorean Theorem: π₯ 2 + π¦ 2 = 1 UC revisited y (cosο± , sin ο± ) 1 ΞΈ sin ο± x cosο± Pythagorean Theorem:πππ 2 π + π ππ2 π = 1 The trig relationships: πππ 2 π + π ππ2 π = 1 πππ 2 π = 1 β π ππ2 π π ππ2 π = 1 β πππ 2 π β’ An identity is an equation that is true for all values of the variables. β’ Difference between identity and equation: β’ An identity is true for any value of the variable, but an equation is not. For example the equation 3x=12 is true only when x=4, so it is an equation, but not an identity. What are identities used for? β’ They are used in simplifying or rearranging algebraic expressions. β’ By definition, the two sides of an identity are interchangeable, so we can replace one with the other at any time. β’ In this section we will study identities with trig functions. The trigonometry identities β’ There are dozens of identities in the field of trigonometry. β’ Many websites list the trig identities. Many websites will also explain why identities are true. i.e. prove the identities. β’ For an example of such a site: click here Trigonometric Identities Quotient Identities sin ο± tan ο± ο½ cos ο± cos ο± cot ο± ο½ sin ο± Reciprocal Identities 1 sin ο± ο½ csc ο± 1 cos ο± ο½ sec ο± 1 tan ο± ο½ cot ο± Pythagorean Identities sin2ο± + cos2ο± = 1 tan2ο± + 1 = sec2ο± cot2ο± + 1 = csc2ο± sin2ο± = 1 - cos2ο± tan2ο± = sec2ο± - 1 cot2ο± = csc2ο± - 1 cos2ο± = 1 - sin2ο± 5.4.3 Where did our pythagorean identities come from?? Do you remember the Unit Circle? β’ What is the equation for the unit circle? x2 + y2 = 1 β’ What does x = ? What does y = ? (in terms of trig functions) sin2ΞΈ + cos2ΞΈ = 1 Pythagorean Identity! Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos2ΞΈ sin2ΞΈ + cos2ΞΈ = 1 . cos2ΞΈ cos2ΞΈ cos2ΞΈ tan2ΞΈ + 1 = sec2ΞΈ Quotient Identity another Pythagorean Identity Reciprocal Identity Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin2ΞΈ sin2ΞΈ + cos2ΞΈ = 1 . sin2ΞΈ sin2ΞΈ sin2ΞΈ 1 + cot2ΞΈ = csc2ΞΈ Quotient Identity a third Pythagorean Identity Reciprocal Identity Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. a) cos ο± ο« sin ο± tan ο± sinο± ο½ cos ο± ο« sin ο± cos ο± sin2 ο± ο½ cos ο± ο« cos ο± cos ο± ο« sin ο± ο½ cos ο± 1 ο½ cos ο± 2 ο½ sec ο± 2 cot 2 ο± 1 ο sin2 ο± b) cos 2 ο± 2 sin ο± ο½ cos 2 ο± 1 cos 2 ο± 1 ο½ ο΄ 2 sin ο± cos2 ο± ο½ 1 2 sin ο± ο½ csc 2 ο± 5.4.5 β’ Practice Problems for Day 1: refer to class handout. While you wait β’ Factor: a) π₯ 2 β 4 b) π₯ 2 β 36 c) π₯ 2 β 1 d) 1 β π₯ 2 β’ Identify as True or False: A. cos βπ = βcos(π) B. sin βπ = βπ ππ(π) C. tan βπ = βtan(π) Proving a Trigonometric Identity: 1. 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. Guidelines for Proving Identities: 1. 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 expression that will be easier to simplify. 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! Prove ππππ¨(π + ππππ π¨) = ππππ π¨ ππππ¨ ππ¨ππ(π¬ππ π π) πππ§π = ππππ π¨ Pythagorean Relationship πππ π΄ 1 ( 2 ) π πππ΄ πππ π΄ π πππ΄ πππ π΄ Definition of trig Functions 2 =ππ π π΄ 1 π πππ΄πππ π΄ = ππ π 2 π΄ π πππ΄ πππ π΄ Reduce πππ π΄ 2 =ππ π π΄ 2 π ππ π΄πππ π΄ 1 2 = ππ π π΄ π ππ2 π΄ 2 2 ππ π π΄ = ππ π π΄ Reduce Def of trig function. Practice Problems Day 2 Sec 8- Written Exercises page 321 #13-19 odds; 29-35 odds Exit Question: #3b the handout. A complete, step by step solution must be included. Using the identities you now know, find the trig value. 1.) If cosΞΈ = 3/4, find secΞΈ 2.) If cosΞΈ = 3/5, find cscΞΈ. sin 2 ο± ο« cos 2 ο± ο½ 1 1 1 4 sec ο± ο½ ο½ ο½ cosο± 3 3 4 2 ο¦ οΆ 3 sin 2 ο± ο« ο§ ο· ο½ 1 ο¨ 5οΈ 25 9 sin 2 ο± ο½ ο 25 25 16 2 sin ο± ο½ 25 4 sin ο± ο½ ο± 5 csc ο± ο½ 1 1 5 ο½ ο½ο± sin ο± ο± 4 4 5 3.) sinΞΈ = -1/3, find tanΞΈ tan 2 ο± ο«1 ο½ sec 2 ο± tan 2 ο± ο«1 ο½ (ο3) 2 tan ο± ο½ 2 2 tan 2 ο± ο½ 8 tan 2 ο± ο½ 8 ο 4.) secΞΈ = -7/5, find sinΞΈ ο Simplifing Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x 1 cos x sin x 2 ο½ 1 ο« 2 tan x ο« tan x ο 2 cos x ο½ (1 ο« tan x) ο 2 sin x 2 ο½ 1 ο« tan2 x ο« 2tanx ο 2 tanx ο½ sec2 x d) csc x tan x ο« cot x 1 ο½ sin x sin x cos x ο« cos x sin x 1 ο½ sin x sin 2 x ο« cos 2 x sin xcos x 1 ο½ sin x 1 sin x cos x 1 sin x cos x ο½ ο΄ sin x 1 ο½ cos x Simplify each expression. 1 ο¦ cos x οΆ cos xο§ ο· ο« sin x ο¨ sin x οΈ sin ο± cos sin ο± ο¦ 1 οΆο¦ sin x οΆ ο½ cos xο§ ο·ο§ ο· ο¨ sin x οΈο¨ cos x οΈ 1 sin ο± ο· sin ο± cos ο± ο½1 ο 1 ο½ sec ο± cosο± cos 2 x sin 2 x ο« sin x sin x cos 2 x ο« sin 2 x sin x 1 ο½ csc x sin x Simplifying trig Identity Example1: simplify tanxcosx sin x tanx cosx cos x tanxcosx = sin x Simplifying trig Identity Example2: simplify sec x csc x 1 cos sec x csc 1x sin x = 1 sinx x cos x 1 = sin x cos x = tan x Simplifying trig Identity Example2: simplify cos2x - sin2x cos x cos2x - sin 1 2x cos x = sec x Example Simplify: = cot x (csc2 x - 1) Factor out cot x = cot x (cot2 x) Use pythagorean identi = cot3 x Simplify Example Simplify: = sin x (sin x) + cos x cos x cos x 2 = sin x + (cos x)cos x cos x = sin2 x + cos2x cos x = 1 cos x = sec x Use quotient identity Simplify fraction with LCD Simplify numerator Use pythagorean iden Use reciprocal identity Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity Practice 1 One way to use identities is to simplify expressions 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. Letβs see an example of this: substitute using each identity sin x tan x ο½ cos x tan x csc x Simplify: sec x simplify sin x 1 ο ο½ cos x sin x 1 cos x 1 ο½ cos x 1 cos x ο½1 1 csc x ο½ sin x 1 sec x ο½ cos x Another way to use identities is to write one function in terms of another function. Letβs see an example of this: Write the following expression in terms of only one trig function: cos x ο« sin x ο« 1 2 = 1 ο sin 2 x ο« sin x ο« 1 This expression 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. = ο sin 2 x ο« sin x ο« 2 sin 2 x ο« cos 2 x ο½ 1 cos 2 x ο½ 1 ο sin 2 x (E) Examples β’ Prove tan(x) cos(x) = sin(x) LS ο½ tan x cos x sin x LS ο½ cos x cos x LS ο½ sin x ο LS ο½ RS 38 (E) Examples β’ Prove tan2(x) = sin2(x) cos-2(x) RS ο½ sin 2 x cos ο2 x 1 οΆ 2ο¦ RS ο½ ο¨sin x ο© ο§ ο· 2 cos x ο¨ οΈ 1 2 RS ο½ ο¨sin x ο© ο¨cos x ο©2 RS ο½ ο¨sin x ο©2 ο¨cos x ο©2 ο¦ sin x οΆ RS ο½ ο§ ο· ο¨ cos x οΈ RS ο½ tan 2 x ο RS ο½ LS 2 39 (E) Examples β’ Prove tan x ο« 1 1 ο½ tan x sin x cos x 1 tan x sin x 1 ο« sin x cos x cos x sin x cos x ο« cos x sin x sin x sin x ο« cos x cos x cos x sin x 2 sin x ο« cos2 x cos x sin x 1 cos x sin x ο½ RS LS ο½ tan x ο« LS ο½ LS ο½ LS ο½ LS ο½ LS ο½ ο LS 40 (E) Examples β’ Prove LS LS LS LS sin 2 x ο½ 1 ο« cos x 1 ο cos x sin 2 x ο½ 1 ο cos x 1 ο cos2 x ο½ 1 ο cos x (1 ο cos x )(1 ο« cos x ) ο½ (1 ο cos x ) ο½ 1 ο« cos x ο LS ο½ RS 41