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) 1)

2) 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 x 1) tan x = (sin x)/(cos x)

2) sec x = 1/cos x 2) 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 x 1) tan x = (sin x)/(cos x)

1) sin (-x) = - sin x

2) sec x = 1/cos x 2) 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 Identities

Tangent and Cotangent Ratio Identities

Pythagorean Identities

Negative-Angle Identities

1) csc x = 1/sin x 1) tan x = (sin x)/(cos x)

1) (sin x)^2 + (cos x)^2 = 1

1) sin (-x) = - sin x

2) sec x = 1/cos x 2) 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 TheoremSohCahToa

Mini Proof

Sin = y/1 ….. Which implies what?Cos = x/1 ….. Which implies what?

Remember that x^2 + y^2 = 1?

So then, using substitution, we know that…

(Sin )^2 + (Cos )^2 = 1

Your Turn!!

Rewrite each expression in terms of cos , and simplify.• A)

• B) sec – (tan )*(sin )

Rewrite each expression in terms of sin , and simplify.• A)

• B)

Proof Time!!

• The second Pythagorean Identity is: 1 + (tan )^2 = (sec )^2

• Prove it using the identities you already know. Hint: Start with the first Pythagorean Identity.

Third Pythagorean Identity

• The second Pythagorean Identity is: (cot )^2 + 1 = (csc )^2

• Prove it using the identities you already know. Hint: Start with the first Pythagorean Identity.

Last Problem!!!

Which is equivalent to 1 – (sec )^2?

•A) (tan )^2•B) -(tan )^2•C) (cot )^2•D) -(cot )^2

Reciprocal Identities

Tangent and Cotangent Ratio Identities

Pythagorean Identities

Negative-Angle Identities

1) csc x = 1/sin x 1) 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 x 2) cot x = (cos x)/(sin x)

2) 1 + (tan )^2 = (sec )^2

2) cos (-x) = cos x

*sec (-x) = sec x

3) cot x = 1/tan x 3) (cot )^2 + 1 = (csc )^2

3) 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.