Transcript Right Triangle Trigonometry

CHAPTER 10
Geometry
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10.6
Right Triangle Trigonometry
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Objectives
1. Use the lengths of the sides of a right triangle
to find trigonometric ratios.
2. Use trigonometric ratios to find missing parts
of right triangles.
3. Use trigonometric ratios to solve applied
problems.
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Ratios in Right Triangles
Trigonometry means measurement of triangles.
Trigonometric Ratios: Let A represent an acute angle of
a right triangle, with right angle, C, shown here.
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Ratios in Right Triangles
For angle A, the trigonometric ratios are
defined as follows:
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Example 1: Becoming Familiar with The
Trigonometric Ratios
Find the sine, cosine, and tangent of A.
Solution: Using the Pythagorean Theorem,
find the measure of the hypotenuse c.
c²  a²  b²  5²  12²  25  144  169
c  169  13
Now apply the trigonometric
ratios: sinA  side opposite angle A 
5
hypotenuse
13
side adjacent to angle A 12
cosA 

hypotenuse
13
side opposite angle A
5
tanA 

side adjacent to angle A 12
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Example 2: Finding a Missing Leg of a Right
Triangle
Find a in the right triangle
Solution: Because we have a
known angle, 40°, with a known
tangent ratio, and an unknown
opposite side, “a,” and a known
adjacent side, 150 cm, we can
use the tangent ratio.
tan 40° = a
150
a = 150 tan 40° ≈ 126 cm
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Applications of the Trigonometric Ratios
• Angle of elevation:
Angle formed by a
horizontal line and the
line of sight to an object
that is above the
horizontal line.
• Angle of depression:
Angle formed by a
horizontal line and the
line of sight to an object
that is below the
horizontal line.
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Example 4: Problem Solving using an
Angle of Elevation
Find the approximate height of this
tower.
Solution: We have a right triangle
with a known angle, 57.2°, an
unknown opposite side, and a
known adjacent side, 125 ft.
Using the tangent ratio:
a
tan 57.2° =
125
a = 125 tan 57.2° ≈ 194 feet
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Example 5: Determining the Angle of Elevation
A building that is 21 meters tall
casts a shadow 25 meters long.
Find the angle of elevation of the
sun.
Solution: We are asked to
find mA.
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Example 5 continued
Use the inverse tangent key
The display should show approximately 40. Thus the
angle of elevation of the sun is approximately 40°.
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