Transcript PC 4-3 Right Triangle Trigx

Chapter 4
Trigonometric Functions
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4.3 Right Triangle Trigonometry
Objectives:
Evaluate trigonometric functions of acute
angles.
Use fundamental trigonometric identities.
Use a calculator to evaluate trigonometric
functions.
Use trigonometric functions to model and
solve real-life problems.
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Right Triangle Definitions of
Trigonometric Functions
Write the six trigonometric functions of angle θ
using the right triangle shown below.
Note that θ is an acute angle.
That is, 0°≤ θ ≤ 90°.
(θ lies in the first quadrant.)
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Example 1
Find the exact values of the six trig functions
of the angle θ shown in the figure.
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Trig Functions of Special Angles
Use special triangles to find the exact values
of the trig functions of angles 45°, 30°, and
60°.
30°
45°
60°
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Summary of Special Angles
Note:
sin
30° = cos 60°
sin 60° = cos 30°
sin 45° = cos 45°
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Cofunctions
Cofunctions of complementary angles
are equal.
sin (90° - θ) = cos θ
tan (90° - θ) = cot θ
sec (90° - θ) = csc θ
cos (90° - θ) = sin θ
cot (90° - θ) = tan θ
csc (90° - θ) = sec θ
Cofunctions are sine & cosine,
tangent & cotangent, secant & cosecant.
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Fundamental Trig Identities
List the six reciprocal identities.
List the two quotient identities.
List the three Pythagorean identities.
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Example 2
Let be θ an acute angle such that sin θ = 0.6.
Use trig identities to find:
a. cos θ
b. tan θ
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Example 3
Use trig identities to transform one side of
the equation into the other (0 < θ < π/2).
a. cos θ sec θ = 1
b. (sec θ + tan θ)(sec θ – tan θ) = 1
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Angles of Elevation and Depression
Angle
of Elevation
The angle from the
horizontal upward to
the object.
Angle of Depression
The angle from the
horizontal downward
to the object.
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Example 4
A surveyor is standing 50 feet from the base
of a large tree, as shown in the figure. The
surveyor measures the angle of elevation to
the top of the tree as 71.5°. How tall is the
tree?
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Example 5
Find the length c of the skateboard ramp
shown in the figure.
18.4°
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Example 6
In traveling across flat land you notice a
mountain directly in front of you. Its angle
of elevation (to the peak) is 3.5°. After you
drive 13 miles closer to the mountain, the
angle of elevation is 9°. Approximate the
height of the mountain.
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Homework 4.3
Worksheet 4.3
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