Download inverse trigonometric ratios

7
THE NATURE OF
GEOMETRY
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7.5
Right-Triangle Trigonometry
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Right-Triangle Trigonometry
An important theorem from geometry, the Pythagorean
theorem, has an important algebraic representation, and is
important in our study of triangles.
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Right-Triangle Trigonometry
A correctly labeled right triangle is shown in Figure 7.49. In
a right triangle, the sides that are not the hypotenuse are
sometimes called legs.
Right triangle
Figure 7.49
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Example 1 – Build a right angle
A carpenter wants to make sure that the corner of a closet
is square (a right angle). If she measures out sides of 3 feet
and 4 feet, how long should she make the diagonal
(hypotenuse)?
Solution:
The length of the hypotenuse is the unknown, so use the
Pythagorean theorem:
The sides are 3 and 4.
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Example 1 – Solution
cont’d
She should make the diagonal 5 feet long.
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Trigonometric Ratios
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Trigonometric Ratios
There are six possible ratios for the triangle shown in
Figure 7.49.
Right triangle
Figure 7.49
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Trigonometric Ratios
The trigonometric ratios, are defined in the box.
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Example 2 – Find angles in a triangle using trigonometry
Given a right triangle with sides of length 5 and 12, find the
trigonometric ratios for the angles A and B. Show your
answers in both common fraction and decimal fraction
form, with decimals rounded to four places.
Solution:
First use the Pythagorean theorem to find the length of the
hypotenuse.
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Example 2 – Solution
cont’d
sin A =
 0.3846; cos A =
 0.9231; tan A =
 0.4167
sin B =
 0.9231; cos B =
 0.3846; tan B =
 2.4
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Example 3 – Exact values for 45° angle
Find the cosine, sine, and tangent of 45°.
Solution:
If one of the angles of a right triangle is 45°, then the other
acute angle must also be 45°(because the sum of the
angles of a triangle is 180°).
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Example 3 – Solution
cont’d
Furthermore, since the base angles have the same
measure, the triangle is isosceles. By the Pythagorean
theorem,
x2 + x2 = r2
2x2 = r2
Note that x is positive.
Now use the definition of the trigonometric ratios.
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Example 3 – Solution
cont’d
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Inverse Trigonometric Ratios
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Inverse Trigonometric Ratios
We can also use right-triangle trigonometry to find one of
the acute angles if we know the trigonometric ratio. For
example, suppose we know (as we do from Example 3)
that
tan  = 1
Also suppose that we do not know the angle . In other
words, we ask, “What is the angle ?” We answer by
saying, “ is the angle whose tangent is 1.” In mathematics,
we call this the inverse tangent and we write
 = tan–11
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Inverse Trigonometric Ratios
To find the angle , we turn to a calculator. Find the button
labeled
and press
The display is 45, which means  = 45°.
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Inverse Trigonometric Ratios
We now define the inverse trigonometric ratios for a right
triangle.
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Example 6 – Find angles of a triangle using inverse trigonometric ratios
Given a right triangle with sides of length 5 and 12, find the
measures of the angles of this triangle.
Solution:
First use the Pythagorean theorem to find the length of the
hypotenuse.
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Example 6 – Solution
cont’d
Also
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Example 6 – Solution
cont’d
Our task here is to find the measures of angles A and B.
What is A? We might say, “A is the measure of the angle
whose sine is ” This is the inverse sine, and we write
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Inverse Trigonometric Ratios
The angle of elevation is the acute angle measured up
from a horizontal line to the line of sight, whereas if we take
the climber’s viewpoint, and measure from a horizontal
down to the line of sight, we call this angle the angle of
depression.
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Example 7 – Find the height of a tree from angle of elevation
The angle of elevation to the top of a tree from a point on
the ground 42 ft from its base is 33°. Find the height of the
tree (to the nearest foot).
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Example 7 – Solution
Let  = angle of elevation and h = height of tree. Then
tan  =
h = 42 tan 33°
 27.28
The tree is 27 ft tall.
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