1. Activate Prior Knowledge CFU
Learning Objective
We will determine1 how to use the Pythagorean Identity to find the trigonometric ratios.
What are we going to do?
What is determine means?_____.
CFU
Activate Prior Knowledge
Pythagoras' theorem is a fundamental relation among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Pythagorean Theorem
On your whiteboard, using the Pythagoras' Theorem find the value of C. Show your steps!
Activate Prior Knowledge CFU
Students, you already know how to use the Pythagorean Theorem. Today, we will learn how to use the Pythagorean Identity to find the trigonometric ratios.
Make Connection
1 Figure out
Vocabulary

2. Pythagorean Identity I II IV III
Concept Development II I
III IV
Pair-Share: A explain to B: How is the Pythagorean theorem similar and/or different from the Pythagorean Identity.
CFU
Pythagorean Identity

3. Concept Development
Whiteboard - - - - - -
Pair-Share: B explain to A: How to solve for Cos.
CFU

4. Use the identity to solve for cos θ.
Skill Development/Guided Practice
The Pythagorean identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
Steps to find Trig Ratio 1 2 3 4
How did I/you know which identity to use?
How did I/you solve for identity?
How did I/you check for sign?
CFU 1 2 3
Identify which Identity to use.
Use a calculator
Pythagorean Identity
Round correctly
Check your signs
1. Given that sin θ = where
find cos θ. If necessary, round to 3 decimal places.
2 . Given that sin θ = −0.442 where
find cos θ. If necessary, round to 3 decimal places.
Since θ lies in Quadrant IV, where cos θ > 0, cos θ
is approximately
Since θ lies in Quadrant I, where cos θ > 0, cos θ
 is approximately:____
Use the identity to solve for cos θ.
Use the identity to solve for cos θ.
Substitute for sin θ .
Substitute for sin θ .
Use a calculator, then round.
Use a calculator, then round. =

5. Use the identity to solve for sin θ.
Skill Development/Guided Practice
The Pythagorean identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
Steps to find Trig Ratio 1 2 3 4
How did I/you know which identity to use?
How did I/you solve for identity?
How did I/you check for sign?
CFU 1 2 3
Identify which Identity to use.
Use a calculator
Pythagorean Identity
Round correctly
Check your signs
1.Given that cos θ = −0.991 where π
find sin θ. Then find tan θ. If necessary, round to
3 decimal places. Since θ lies in Quadrant II,
where sin θ > 0, sin θ ≈ ___ and tan θ ≈ ___
2.Given that cos θ = −0.996 where π
find sin θ. Then find tan θ. If necessary, round to
3 decimal places. Since θ lies in Quadrant II,
where sin θ > 0, sin θ ≈ ___ and tan θ ≈ ___
Use the identity to solve for sin θ.
Use the identity to solve for sin θ.
Substitute for cos θ .
Substitute for cos θ
Use a calculator, then round.
Use a calculator, then round.
Find Tan,
Find Tan,

6. The tree is about 76 feet tall. Write the ratio Substitute values
Relevance Reason #1: Trig Ratio are used in finding the height.
You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.
tan 59° =
opposite
adjacent
Write the ratio
tan 59° = h 45
The tree is about
76 feet tall.
Check for Understanding
Does anyone else have another reason why it is relevant to use verb tense correctly?
Which reason is most relevant to you? Why?
Substitute values
45 tan 59° = h
Multiply each side by 45
45 (1.6643) ≈ h
Sample Item
75.9 ≈ h
Simplify
Find Sin, Cos, Tan of T. Leave answer as a fraction.
Relevance Reason #2: Know how to find Trig Ratios will help you do well on tests: (PSAT, SAT, ACT, GRE, GMAT, LSAT, etc..).

7. What did you learn today about how to use the Pythagorean Identity to find the trigonometric ratios.
Word Bank
Unit Circle
Pythagorean Identity
Pythagorean Theorem .
SUMMARY CLOSURE
Today, I learned how to __________________
______________________________________________________________.