Download GEOMETRY MODULE 2 LESSON 27 SINE AND COSINE OF

GEOMETRY
MODULE 2 LESSON 27
SINE AND COSINE OF COMPLEMENTARY ANGLES
AND SPECIAL ANGLES
OPENING EXERCISE
Complete Example 1 from your workbook.

Why do we know that alpha, 𝛼 and beta, 𝛽 are complementary?
The sum of all the angles in a triangle is 180°. Since angle C is the right
angle, the sum of 𝛼 and 𝛽 must be 90°.
WORKBOOK
Exercise 1 (Together)
Consider the right triangle ABC so that ∠𝐶 is a right angle, and the degree
measures of ∠𝐴 and ∠𝐵 are 𝛼 and 𝛽, respectively.
a. Find 𝛼 + 𝛽.
𝛼 + 𝛽 = 90°
𝐵𝐶
b. Use trigonometric ratios to describe 𝐴𝐵 in two different ways.
𝐵𝐶
𝐵𝐶
sin ∠𝐴 = 𝐴𝐵 , cos ∠𝐵 = 𝐴𝐵
c. Use trigonometric ratios to describe
𝐴𝐶
𝐴𝐶
𝐴𝐵
in two different ways.
𝐴𝐶
sin ∠𝐵 = 𝐴𝐵 , cos ∠𝐴 = 𝐴𝐵
d. What can you conclude about sin 𝛼 and cos 𝛽?
sin 𝛼 = cos 𝛽
e. What can you conclude about cos 𝛼 and sin 𝛽?
cos 𝛼 = sin 𝛽
MOD2 L27
1
ON YOUR OWN
Complete Exercise 2 and 3 in your workbook.
2. a) 𝜃 = 65 b) 𝜃 = 10 c) 𝜃 = 40 d) 𝜃 = 67.5
3. Sine and cosine have the same value for 𝜃 = 45. The sine of an angle is equal to the cosine of its
complement. Since the complement of 45 is 45, sin 45 = cos 45.
DISCUSSION
http://www.rkm.com.au/ANIMATIONS/animation-Pythagoras-Theorem.html

What is happening to 𝑎 and 𝑏 as 𝜃 changes?
What happens to sin 𝜃 and cos 𝜃? Let 𝑐 = 1.
o As 𝜃 gets smaller approaching 0°,
𝑎
𝑎 decreases. Since sin 𝜃 = 1, sin 𝜃 is approaching 0.
𝑏
𝑏 increases. Since cos 𝜃 = 1, sin 𝜃 is approaching 1.
o As 𝜃 gets bigger approaching 90°,
𝑎
𝑎 increases. Since sin 𝜃 = 1, sin 𝜃 is approaching 1.
𝑏
𝑏 decreases. Since cos 𝜃 = 1, sin 𝜃 is approaching 0.
EXAMPLE 3
a.
What’s an easy way to remember the entries in the table?
1 √2 √3
, ,1
2 2 2
Consider the sequence 0, ,
or
√0 1 √2 √3 √4
, , , , .
2 2 2 2 2
Apply the sequence left to right in the sine
row then right to left in the cosine row.
𝜽
Sine
Cosine
MOD2 L27
2
THE UNIT TRIANGLES
Create a similar triangle. Choose a scale factor and apply through multiplication.
Complete exercise 4 and 5 in your workbook.
MOD2 L27
3
SUMMARY

The sine of an angle is equal to the cosine of its complementary angle, and the cosine of an angle is
equal to the sine of its complementary angle.

Sine and Cosine for Special Angles:
HOMEWORK
Problem Set Module 2 Lesson 27, page 212
#1, 3, 4, 5: 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 17, 2017
MOD2 L27
4