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Example 4: The following diagrams show the basic side lengths of the 30-60-90
and 45-45-90 triangles.
600
450
2
1
1
300
450
1
Determine side lengths of three different triangles similar to each of the
triangle in the chart. Then generalize the side lengths in terms of x.
30­60­90 Triangle
side lengths 1: : 2
45­45­90 Triangle
side lengths 1: 1 : ______ : ______ : ______
______ : ______ : ______
______ : ______ : ______
______ : ______ : ______
______ : ______ : ______
______ : ______ : ______
x : ______ : ______
x : ______ : ______
Exercises
1. The triangles below are 30-60-90 right triangles. Find the unknown lengths a
and b, using sin and cos values of an acute angle. Show your solving of equations.
a)
b)
3
a
c
3
300
b
600
a
c)
e)
d)
a
b
45
c
450
a
c
a
0
450
2. Given an equilateral triangle with sides of length 9, find the length of the
altitude.
Confirm your answer using the Pythagorean Theorem.
Let's Sum it Up!
• The sine of an angle is equal to the cosine of its complementary angle, and the
cosine of and angle is equal to the sine of its complementary angle.
• Sin 900 = 1 and cos 00 = 1 and similarly, sin 00 = 0 and cosine of 900 = 1.
• The values for the cosine and sine values for the special angles are the same,
but they are in reverse order.
Name_____________________
Date _____________________
CC Geometry H
HW #27
#1-6 Find the values of θ that make the equation true.
1. sin θ = cos 32
2. cos 11 = sin θ
3. sin θ = cos (θ + 38)
3. sin (θ + 10) = sin 60
4. cos θ = sin (3θ + 20)
6.
#7-12 The triangles below are 30-60-90 right triangles. Find the unknown
lengths x and y, using sin and cos values of an acute angle. Show your solving of an
appropriate equation.
9.
8.
7.
y
300
600
600
y
7
x
x
x
y
12
11.
10.
12.
10
x
450
y
450
x
x
45
0
y
y
OVER
13. A square has side lengths 7√2. Use sine or cosine to find the length of the
diagonal of the square. Confirm your answer using the Pythagorean Theorem.
7√2
7√2
14a) Make a prediction about how the sum sin 30 + cos 60 will relate to the
sum sin 60 + cos 30.
b) Use the sine and cosine values of special angles to find the sum: sin 30 + cos 60.
c) Find the exact value of the sum: sin 60 + cos 30.
d) Was your prediction correct ? Explain why or why not.