Download Module 2 Lesson 30 with Notes

GEOMETRY
MODULE 2 LESSON 30
TRIGONOMETRY AND THE PYTHAGOREAN THEOREM
OPENING EXERCISE
Complete Exercise 1 and 2 from your workbook.
DISCUSSION
We will derive the Pythagorean Identity using the Pythagorean Theorem:

State the Pythagorean Theorem using leg and hypotenuse.
𝑙𝑒𝑔2 + 𝑙𝑒𝑔2 = ℎ𝑦𝑝2

Substitute leg for opposite and adjacent,
𝑜𝑝𝑝2 + 𝑎𝑑𝑗 2 = ℎ𝑦𝑝2

Divide both sides by ℎ𝑦𝑝2 .
𝑜𝑝𝑝2 𝑎𝑑𝑗 2 ℎ𝑦𝑝2
+
=
ℎ𝑦𝑝2 ℎ𝑦𝑝2 ℎ𝑦𝑝2

Make a substitution with trigonometric functions.
(sin 𝜃)2 + (cos 𝜃)2 = 1
The Pythagorean identity can also be written as sin2 𝜃 + cos2 𝜃 = 1 on the Web and other
textbooks. In Precalculus and Advanced Topics, this notation is used. You may use either.
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1
ON YOUR OWN
Complete Example 2 in your workbook. Hint: Substitute the ratios for sine and cosine.
sin 𝜃
tan 𝜃 = cos 𝜃
Using the information from the given right triangle:
sin 𝜃 =
𝑎
𝑐
and cos 𝜃 =
𝑏
𝑐
Isolating a and b yields:
𝑐 ∙ sin 𝜃 = 𝑎 and 𝑐 ∙ cos 𝜃 = 𝑏
Using the information from the given right triangle:
tan 𝜃 =
𝑎
𝑏
Substituting a and b:
tan 𝜃 =
𝑐 ∙ sin 𝜃
𝑐 ∙ cos 𝜃
Simplify:
tan 𝜃 =
sin 𝜃
cos 𝜃
WORKBOOK
1
Exercise 3: In a right triangle with acute angle of measure 𝜃, sin 𝜃 = 2. Use the Pythagorean Identity
to determine the value of cos 𝜃.
sin2 𝜃 + cos2 𝜃 = 1
1 2
( ) + cos2 𝜃 = 1
2
1
+ cos 2 𝜃 = 1
4
1
cos 2 𝜃 = 1 −
4
3
cos 2 𝜃 =
4
3 √3
cos 𝜃 = √ =
4 √4
cos 𝜃 =
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√3
2
2
ON YOUR OWN
Complete Exercise 4 in your workbook. Hint: Find cos 𝜃 first.
sin2 𝜃 + cos2 𝜃 = 1
7 2
( ) + cos 2 𝜃 = 1
9
49
+ cos2 𝜃 = 1
81
49
cos2 𝜃 = 1 −
81
32
cos 2 𝜃 =
81
cos 𝜃 = √
32 4√2
=
81
9
7
sin 𝜃
tan 𝜃 =
= 9
cos 𝜃 4√2
9
7
9
7
tan 𝜃 = ×
=
9 4√2 4√2
tan 𝜃 =
7√2
8
SUMMARY

The Pythagorean Identity: sin2 𝜃 + cos2 𝜃 = 1

Representations of Tangent: tan 𝜃 =

If one of the values sin 𝜃 , cos 𝜃, or tan 𝜃 is provided to us, we can find the other two values by
𝑜𝑝𝑝
𝑎𝑑𝑗
sin 𝜃
= cos 𝜃
sin 𝜃
using the identities sin2 𝜃 + cos2 𝜃 = 1, tan 𝜃 = cos 𝜃 or the Pythagorean Theorem.
HOMEWORK
Problem Set Module 2 Lesson 30, page 228
#1 thru #6 and #8
Show all work in an organized and linear manner. There is not enough room in the workbook. Present
your homework on a separate sheet of paper.
DUE: Tuesday, Jan 24, 2017
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