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9-12 Integrated Math 3 Unit 6 Lesson 6

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Lesson 6

6.1 Warm-up 6.2 Activity 6.3 Activity Student Lesson Summary

Unit 6, Lesson 6

The Pythagorean Identity (Part 2)

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Integrated Math 3

6.1

Warm-up

For an angle in the quadrant indicated, use mental estimation to identify the values of , , and as either positive or negative.

Quadrant I
Quadrant II
Quadrant III
Quadrant IV

6.2

Activity

Suppose that the angle , in radians, is in Quadrant IV of the unit circle. If , what are the values of and ?

Andre uses the Pythagorean Identity and determines that the value of is -0.96. Using the values of sine and cosine, he then calculates the value of tangent:

Do you agree with Andre? Explain or show your reasoning.

6.3

Activity

Your teacher will give you a set of cards that should be arranged face up with cards showing values for sine, cosine, and tangent on one side and cards showing quadrants on the other.

Take turns with your partner matching pairs of cards. Identify 2 pairs that are possible on the unit circle and 2 pairs that are not possible, in any order.
1. For each pair, explain to your partner how you know if the pair is or is not possible on the unit circle. Once a pair is identified, place the cards in front of you to use later.
2. For each pair that your partner draws, listen carefully to the explanation. If you disagree, discuss your thinking and work to reach an agreement.
Once you and your partner have each identified 4 pairs, pick 1 of your possible matches and then calculate the values of the two missing trigonometric ratios for that match. When you finish, trade calculations with your partner and check each other's work. If you disagree, discuss your thinking and work to reach an agreement.

Student Lesson Summary

Suppose we know that and that is an angle in Quadrant III. What can we say about the values of cosine and tangent at ?

Since we can think of as the -coordinate of a point in Quadrant III, let’s start with a sketch of the unit circle showing point .

<p>Circle on the coordinate plane, center at the origin. Point P on the circle in the third quadrant, central angle counter-clockwise from positive x axis has measure theta.</p>

The sketch helps us see that the -coordinate, which is , is also negative. Using the Pythagorean Identity, we can calculate the value of :

Now that we know the value of cosine, we can calculate the value of tangent with some division:

We can use one piece of information and the structure of the unit circle to figure out a whole bunch more, similar to how we used the value of one side length, the hypotenuse, and the structure of right triangles in the past to figure out the other side length of the right triangle.

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