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Trigonometric Equations Reciprocal and Pythagorean Identities.

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Presentation on theme: "Trigonometric Equations Reciprocal and Pythagorean Identities."— Presentation transcript:

1 Trigonometric Equations Reciprocal and Pythagorean Identities

2 DO NOW 1) Take out Homework from Ms. Chung 2) Take out Paper and Pencils 3) Do Warm Up Warm Up A) Find the reciprocal of 2, -4, pi, x and (-1/2) ? B) What is the formula used for Pythagorean Theorem?

3 Introduction Goals: Trigonometric Equations and Intro to Limits Expectations: Etiquette for Talking, Being on Time, Asking for Help, Note Taking Office Hours: This Week Only: Thursday 3:30-4:30 Next Week: Wednesday 3:30-4:30 HW Policy: HW will be assigned at least once a week

4 AGENDA 1) Reciprocal and Pythagorean Identities 2) Math Fair 3) HW WARNING: Excuse my notation!!

5 What is an Identity? Definition of Identity: An equation that is true for all values of the variables. Examples: 2x = 2x (a-b)(a+b) = a^2 +2ab + b^2 5(x+13) = 5x + 65 Non-examples: 3x + 2 = x 5(y-2) = 2y

6 Your Turn Create the following and fill in the first two columns. Tip: You know these from when you first studied Trig functions. Reciprocal Identities Tangent and Cotangent Ratio Identities Pythagorean Identities Negative-Angle Identities 1) 2) 3)

7 Prove each Trigonometric Identity. A) sec x = (csc x)*(tan x) B) (sin x)*(cot x) = cos x Write an equivalent expression for (sec x)*(sin x) Reciprocal Identities Tangent and Cotangent Ratio Identities Pythagorean Identities Negative-Angle Identities 1) csc x = 1/sin x1) tan x = (sin x)/(cos x) 2) sec x = 1/cos x2) cot x = (cos x)/(sin x) 3) cot x = 1/tan x

8 Check for Understanding Think-Write-Pair-Share Define what is an identity? What is an example? What is a non-example? Why is this important?

9 Do You See Any Patterns?

10 What are the Negative-Angle Identities? Reciprocal Identities Tangent and Cotangent Ratio Identities Pythagorean Identities Negative-Angle Identities 1) csc x = 1/sin x1) tan x = (sin x)/(cos x) 1) sin (-x) = - sin x 2) sec x = 1/cos x2) cot x = (cos x)/(sin x) 2) cos (-x) = cos x 3) cot x = 1/tan x 3) tan (-x) = - tan x

11 Your Turn Prove each trigonometric identity. A) csc (-x) = - csc (x) B) 1 – sec (-x) = 1 – sec (x)

12 Almost There!!!!

13 Reciprocal IdentitiesTangent and Cotangent Ratio Identities Pythagorean IdentitiesNegative-Angle Identities 1) csc x = 1/sin x1) tan x = (sin x)/(cos x) 1) (sin x)^2 + (cos x)^2 = 1 1) sin (-x) = - sin x 2) sec x = 1/cos x2) cot x = (cos x)/(sin x) 2) 2) cos (-x) = cos x 3) cot x = 1/tan x 3) 3) tan (-x) = - tan x

14 Proof of the Pythagorean Theorem SohCahToa

15 Your Turn!!

16 Proof Time!!

17 Third Pythagorean Identity

18 Last Problem!!!

19 Reciprocal Identities Tangent and Cotangent Ratio Identities Pythagorean IdentitiesNegative-Angle Identities 1) csc x = 1/sin x1) tan x = (sin x)/(cos x) 1) (sin x)^2 + (cos x)^2 = 1 1)sin (-x) = - sin x * csc (-x) = - csc x 2) sec x = 1/cos x2) cot x = (cos x)/(sin x)2) cos (-x) = cos x *sec (-x) = sec x 3) cot x = 1/tan x3) tan (-x) = - tan x *cot (-x) = - cot (x)

20 HW Review pg. 459, specifically the table of identities. Read pg. 463, Guidelines for Establishing Identities. Do brain exercises: 9, 11, 13, 19, 23, 27, 49, 53 and 69 on page 464-465 Note: For these 10 exercises, show all steps and justify each step. Due: This Friday, April 25.


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