Download solve a triangle

2
Acute Angles
and Right
Triangles
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Acute Angles and
2
Right Triangles
2.1 Trigonometric Functions of Acute Angles
2.2 Trigonometric Functions of Non-Acute
Angles
2.3 Finding Trigonometric Function Values
Using a Calculator
2.4 Solving Right Triangles
2.5 Further Applications of Right Triangles
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2.4
Solving Right Triangles
Significant Digits ▪ Solving Triangles ▪ Angles of Elevation or
Depression
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Significant Digits
A significant digit is a digit obtained by actual
measurement.
The significant digits in the following numbers are
identified in color.
408
21.5 18.00
6.700 0.0025 0.09810
7300
Your answer is no more accurate than the least
accurate number in your calculation.
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To determine the number of significant digits for
answers in applications of angle measure, use the
following table.
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Solving Triangles
To solve a triangle means to find the measures of all
the angles and sides of the triangle.
When solving triangles, a
labeled sketch is an important
aid.
Use a to represent the length
of the side opposite angle A,
b for the length of the side
opposite angle B, and so on.
In a right triangle, the letter c is reserved for the
hypotenuse.
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Example 1
SOLVING A RIGHT TRIANGLE GIVEN
AN ANGLE AND A SIDE
Solve right triangle ABC, if A = 34°30′ and c = 12.7 in.
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Example 1
SOLVING A RIGHT TRIANGLE GIVEN
AN ANGLE AND A SIDE (continued)
We could have found the measure of angle B first and
then used the trigonometric function values of B to
find the unknown sides.
A right triangle can usually be solved in several ways,
each producing the correct answer.
To maintain accuracy, always use given
information as much as possible, and avoid
rounding off in intermediate steps.
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Example 2
SOLVING A RIGHT TRIANGLE GIVEN
TWO SIDES
Solve right triangle ABC,
if a = 29.43 cm and
c = 53.58 cm.
or 33º 19΄
B  90  3319'  5641'
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Angles of Elevation or Depression
The angle of
elevation from point X
to point Y (above X) is
the acute angle formed
by ray XY and a
horizontal ray with
endpoint at X.
The angle of
depression from point
X to point Y (below X)
is the acute angle
formed by ray XY and
a horizontal ray with
endpoint X.
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Caution
Be careful when interpreting the angle
of depression.
Both the angle of elevation and the
angle of depression are measured
between the line of sight and a
horizontal line.
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Solving an Applied Trigonometry
Problem
Step 1 Draw a sketch, and label it with the
given information. Label the
quantity to be found with a
variable.
Step 2 Use the sketch to write an
equation relating the given
quantities to the variable.
Step 3 Solve the equation, and check that
your answer makes sense.
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Example 3
FINDING A LENGTH GIVEN THE ANGLE
OF ELEVATION
Pat Porterfield knows that when she stands 123 ft from
the base of a flagpole, the angle of elevation to the top
of the flagpole is 26°40′. If her eyes are 5.30 ft above
the ground, find the height of the flagpole.
Since Pat’s eyes are 5.30 ft above the ground, the
height of the flagpole is 61.8 + 5.30 = 67.1 ft.
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Example 4
FINDING AN ANGLE OF DEPRESSION
From the top of a 210-ft cliff,
David observes a lighthouse
that is 430 ft offshore. Find
the angle of depression
from the top of the cliff to
the base of the lighthouse.
The angle of depression is measured from a
horizontal line down to the base of the lighthouse.
The angle of depression and angle B, in the right
triangle shown, are alternate interior angles whose
measures are equal. We use the tangent ratio to
solve for angle B.
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Example 4
FINDING AN ANGLE OF DEPRESSION
(continued)
210
1 210
tan B 
, so B  tan
430
430
 26 Angle of depression
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