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Unit 5 Analytic Trigonometry Homework

Presentation on theme: "Chapter 5: Analytic Trigonometry"— Presentation transcript:

1 Chapter 5: Analytic Trigonometry
Section 5.1a: Fundamental IdentitiesHW: p odd, odd

2 Is this statement true?This identity is a true sentence, but onlywith the qualification that x must be in thedomain of both expressions.If either side of the equality is undefined (i.e., at x = –1), thenthe entire expression is meaningless!!!The statement is a trigonometric identitybecause it is true for all values of the variable for which bothsides of the equation are defined.The set of all such values is called the domain of validity ofthe identity.

3 Basic Trigonometric Identities
Reciprocal IdentitiesQuotient Identitiesis in the domain of validity of exactly three of the basicidentities. Which three?

4 Basic Trigonometric Identities
Reciprocal IdentitiesQuotient IdentitiesFor exactly two of the basic identities, one side of the equationis defined at and the other side is not. Which two?

5 Basic Trigonometric Identities
Reciprocal IdentitiesQuotient IdentitiesFor exactly three of the basic identities, both sides of theequation are undefined at Which three?

6 Pythagorean Identities
Recall our unit circle:PWhat are the coordinates of P?sint(1,0)costSo by the Pythagorean Theorem:Divide by :

7 Pythagorean Identities
Recall our unit circle:PWhat are the coordinates of P?sint(1,0)costSo by the Pythagorean Theorem:Divide by :

8 Pythagorean Identities
Given and , find andWe only take the positive answer…why?

9 Cofunction Identities
Can you explain why each of these is true???

10 Odd-Even IdentitiesIf , findSine is odd Cofunction Identity 

11 Simplifying Trigonometric Expressions
Simplify the given expression.How can we support this answer graphically???

12 Simplifying Trigonometric Expressions
Simplify the given expression.Graphical support?

13 Simplifying Trigonometric Expressions
Simplify the given expressions to either a constant or a basictrigonometric function. Support your result graphically.

14 Simplifying Trigonometric Expressions
Simplify the given expressions to either a constant or a basictrigonometric function. Support your result graphically.

15 Simplifying Trigonometric Expressions
Use the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.

16 Simplifying Trigonometric Expressions
Use the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.

17 Simplifying Trigonometric Expressions
Use the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.

18 Let’s start with a practice problem…
Simplify the expressionHow about somegraphical support?

19 Combine the fractions and simplify to a multiple of a power of a
basic trigonometric function.

20 Combine the fractions and simplify to a multiple of a power of a
basic trigonometric function.

21 Combine the fractions and simplify to a multiple of a power of a
basic trigonometric function.

22 Quick check of your algebra skills!!!
Factor the following expression (without any guessing!!!)What two numbers have a product of –180 and a sum of 8?Rewrite middle term:Group terms and factor:Divide out common factor:

23 Write each expression in factored form as an algebraic
expression of a single trigonometric function.e.g.,LetSubstitute:Factor:“Re”substitute for your answer:

24 Write each expression in factored form as an algebraic
expression of a single trigonometric function.e.g.,

25 Write each expression in factored form as an algebraic
expression of a single trigonometric function.e.g.,Let

26 Write each expression in factored form as an algebraic
expression of a single trigonometric function.e.g.,

27 Write each expression as an algebraic expression of a single
trigonometric function.e.g.,

5. Analytical Trigonometry - 1 - www.mastermathmentor.com- Stu SchwartzUnit 5 – Analytical Trigonometry – ClassworkA) Verifying Trig Identities: Definitions to know: Equality: a statement that is always true. example: 2 = 2, 3 + 4 = 7, 62=36, 2 3+5( )=6+10. Equation: a statement that is conditionally true, depending on the value of a variable. example: 2x+3=11, x"1( )2=25, x3"2x2+5x"12=0, 2sin"=1.Identity: a statement that is always true no matter the value of the variable. example: 2x+3x=5x, 4x"3( )=4x"12, x"1( )2=x2"2x+1, 1x"1"1x+1=2x2"1. In the last example, it could be argued that this is not an identity, because it is not true for all values of the variable (xcannot be 1 or -1). However, when such statements are written, we assume the domain is taken into consideration although we don’t always write it. So a better definition of an identity is: a statement that is always true for all values of the variable within its domain. The 8 Fundamental Trigonometric Identities: Trig Identities proofs (assuming in standard position) Reciprocal Identitiescsc=1sinsec=1coscot=1tanQuotient Identitiestan=sincoscot=cossinPythagorean Identitiessin2+cos2=11+tan2=sec21+cot2=csc21sin=1yr=ry=csc1cos=1xr=rx=sec1tan=1yx=xy=cotsincos=yrxr=yx=tancossin=xryr=xy=cotx2+y2=r2x2+y2=r2x2+y2=r2x2r2+y2r2=r2r2x2x2+y2x2=r2x2x2y2+y2y2=r2y2cos2+sin2=1 | 1+tan2=sec2| cot2+1=csc2Corollaries: a statement that is true because another statement is true: Examples (you write the others): Reciprocal identities: sincsc=1 sin=1cscsincos=1Quotient identities: tancos=sincos=sintanPythagorean identities: sin2=1#cos2cos2=1#sin2sin= ±#cos2cos= ±#sin2