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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) 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.
