13.1 Use Trigonometry with
Right Triangles
Sine, cosine, tangent, cosecant, secant, cotangent
The six trigonometric
functions that consist of ratios of a right triangle’s side lengths
RIGHT TRIANGLE DEFINITIONS
OF TRIGONOMETRIC FUNCTIONS
Let q be an acute
angle of a right triangle. The six trigonometric functions of q are defined
as follows:
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cot q = |
The abbreviations opp,
adj, and hyp
are often used to represent the side lengths of the right triangle. Note that
the ratios in the second column are reciprocals of the ratios in the first
column:
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csc q = |
sec q = |
cot q = |
Your Notes
Example 1
Evaluate trigonometric
functions
Evaluate the six
trigonometric functions of the angle 0.
From the Pythagorean theorem, the length of the hypotenuse is
= _17_.
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Sin q = |
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=
TRIGONOMETRIC VALUES FOR
SPECIAL ANGLES
The
table below gives the values of the six trigonometric functions for the angles
30°, 45°, and 60°. You can obtain these values from the triangles shown.
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0 |
sin q |
cos q |
tan q |
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30° |
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45° |
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1 |
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60° |
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q |
csc q |
sec
q |
cot
q |
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30° |
2 |
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45° |
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1 |
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60° |
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2 |
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Example 2
Use a calculator to
solve a right triangle
Solve DABC.
Solution
A
and B are complemetary angels, so A = 90° - _56_ = _34°_
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tan 56° = |
sec 56° = |
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__13(tan 56°)__ = b |
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_19.27_ » b |
_23.25_
» c |
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So, A = _34°_, b » _19.27_, and c » _23.25_. |
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1.
Solve
DABC.
B = 53°, a » 13.24, and b » 17.57
Example 3
Use an angle of
elevation
Building Height You are measuring the height of your school building. You stand
25 feet from the base of the school. The angle of elevation from a point on the
ground to the top of the school is 62°. Estimate the height of the school to
the nearest foot.
Solution
1.
Draw
a diagram that represents the situation.
2.
Write and solve an equation to find the height h.
tan _62°_ =
_25_(tan
62°) = h
_47_
» h
The
height of the school is about _47_ feet.