Trigonometric Equations Reciprocal and Pythagorean Identities.

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Trigonometric Equations Reciprocal and Pythagorean Identities

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?

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

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

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

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)

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

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?

Do You See Any Patterns?

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

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

Almost There!!!!

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

Proof of the Pythagorean Theorem SohCahToa

Proof Time!!

Third Pythagorean Identity

Last Problem!!!

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)

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.