Right-Angle Trigonometry

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10-1
Right-Angle Trigonometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
Holt McDougal Algebra 2
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Warm Up Given the measure of one of the acute
angles in a right triangle, find the measure of
the other acute angle. 1. 45 2. 60 3.
24 4. 38
30
45
66
52
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Warm Up Continued Find the unknown length for
each right triangle with legs a and b and
hypotenuse c. 5. b 12, c 13 6. a 3, b
3
a 5
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Objectives
Understand and use trigonometric relationships of
acute angles in triangles. Determine side
lengths of right triangles by using trigonometric
functions.
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Vocabulary
trigonometric function sine cosine tangent cosecan
ts secant cotangent
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A trigonometric function is a function whose rule
is given by a trigonometric ratio. A
trigonometric ratio compares the lengths of two
sides of a right triangle. The Greek letter theta
? is traditionally used to represent the measure
of an acute angle in a right triangle. The values
of trigonometric ratios depend upon ?.
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The triangle shown at right is similar to the one
in the table because their corresponding angles
are congruent. No matter which triangle is used,
the value of sin ? is the same. The values of the
sine and other trigonometric functions depend
only on angle ? and not on the size of the
triangle.
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Example 1 Finding Trigonometric Ratios
Find the value of the sine, cosine, and tangent
functions for ?.
sin ?
cos ?
tan ?
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Check It Out! Example 1
Find the value of the sine, cosine, and tangent
functions for ?.
sin ?
cos ?
tan ?
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You will frequently need to determine the value
of trigonometric ratios for 30,60, and 45
angles as you solve trigonometry problems. Recall
from geometry that in a 30-60-90 triangle, the
ration of the side lengths is 1 3 2, and that
in a 45-45-90 triangle, the ratio of the side
lengths is 11 2.
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Example 2 Finding Side Lengths of Special Right
Triangles
Use a trigonometric function to find the value of
x.
The sine function relates the opposite leg and
the hypotenuse.
x 37
Multiply both sides by 74 to solve for x.
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Check It Out! Example 2
Use a trigonometric function to find the value of
x.
The sine function relates the opposite leg and
the hypotenuse.
Multiply both sides by 20 to solve for x.
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Example 3 Sports Application
In a waterskiing competition, a jump ramp has
the measurements shown. To the nearest foot,
what is the height h above water that a skier
leaves the ramp?
Substitute 15.1 for ?, h for opp., and 19 for
hyp.
Multiply both sides by 19.
Use a calculator to simplify.
5 h
The height above the water is about 5 ft.
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Check It Out! Example 3
A skateboard ramp will have a height of 12 in.,
and the angle between the ramp and the ground
will be 17. To the nearest inch, what will be
the length l of the ramp?
Substitute 17 for ?, l for hyp., and 12 for opp.
Multiply both sides by l and divide by sin 17.
Use a calculator to simplify.
l 41
The length of the ramp is about 41 in.
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When an object is above or below another object,
you can find distances indirectly by using the
angle of elevation or the angle of depression
between the objects.
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Example 4 Geology Application
A biologist whose eye level is 6 ft above the
ground measures the angle of elevation to the top
of a tree to be 38.7. If the biologist is
standing 180 ft from the trees base, what is the
height of the tree to the nearest foot?
Step 1 Draw and label a diagram to represent the
information given in the problem.
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Example 4 Continued
Step 2 Let x represent the height of the tree
compared with the biologists eye level.
Determine the value of x.
Use the tangent function.
Substitute 38.7 for ?, x for opp., and 180 for
adj.
180(tan 38.7) x
Multiply both sides by 180.
144 x
Use a calculator to solve for x.
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Example 4 Continued
Step 3 Determine the overall height of the tree.
x 6 144 6
150
The height of the tree is about 150 ft.
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Check It Out! Example 4
A surveyor whose eye level is 6 ft above the
ground measures the angle of elevation to the top
of the highest hill on a roller coaster to be
60.7. If the surveyor is standing 120 ft from
the hills base, what is the height of the hill
to the nearest foot?
Step 1 Draw and label a diagram to represent the
information given in the problem.
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Check It Out! Example 4 Continued
Step 2 Let x represent the height of the hill
compared with the surveyors eye level. Determine
the value of x.
Use the tangent function.
Substitute 60.7 for ?, x for opp., and 120 for
adj.
120(tan 60.7) x
Multiply both sides by 120.
214 x
Use a calculator to solve for x.
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Check It Out! Example 4 Continued
Step 3 Determine the overall height of the roller
coaster hill.
x 6 214 6
220
The height of the hill is about 220 ft.
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The reciprocals of the sine, cosine, and tangent
ratios are also trigonometric ratios. They are
trigonometric functions, cosecant, secant, and
cotangent.
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Example 5 Finding All Trigonometric Functions
Find the values of the six trigonometric
functions for ?.
Step 1 Find the length of the hypotenuse.
Pythagorean Theorem.
a2 b2 c2
Substitute 24 for a and 70 for b.
c2 242 702
c2 5476
Simplify.
Solve for c. Eliminate the negative solution.
c 74
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Example 5 Continued
Step 2 Find the function values.
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Check It Out! Example 5
Find the values of the six trigonometric
functions for ?.
Step 1 Find the length of the hypotenuse.
Pythagorean Theorem.
a2 b2 c2
Substitute 18 for a and 80 for b.
c2 182 802
c2 6724
Simplify.
Solve for c. Eliminate the negative solution.
c 82
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Check It Out! Example 5 Continued
Step 2 Find the function values.
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Lesson Quiz Part I
Solve each equation. Check your answer.
1. Find the values of the six trigonometric
functions for ?.
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Lesson Quiz Part II
2. Use a trigonometric function to find the value
of x.
3. A helicopters altitude is 4500 ft, and a
planes altitude is 12,000 ft. If the angle of
depression from the plane to the helicopter is
27.6, what is the distance between the two, to
the nearest hundred feet?
16,200 ft