Right-Angle Trigonometry

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Right-Angle Trigonometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt 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
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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
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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 ?.
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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
ratio 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.
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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?
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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?
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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.
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Check It Out! Example 4 Continued
Step 3 Determine the overall height of the roller
coaster hill.
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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 ?.