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Example 1 Find Trigonometric Values
Example 2 Use One Trigonometric Ratio to Find Another
Example 3 Find a Missing Side Length of a
Right Triangle
Example 4 Solve a Right Triangle
Example 5 Find Missing Angle Measures of Right
Triangles
Example 6 Indirect Measurement
Example 7 Use an Angle of Elevation
Right Triangles
Consider the right triangle
ABC in which the
measure of acute angle A
is identified by the Greek
letter theta, θ. The sides
of the triangle are the
hypotenuse, the leg
opposite θ, and the leg
adjacent to θ.
Right Triangles
Using these sides, you
can define six
trigonometric functions:
Sine
(sin)
Cosine
(cos)
Tangent
(tan)
Secant
(sec)
Cosecant
(csc)
Cotangent
(cot)
How can YOU remember SOHCAHTOA?
Special Right Triangles
45 – 45 – 90
30 – 60 – 90
Introduction to Trigonometric
Ratios with Special Right Triangles
Packet
Pages 2 – 4
Multiple-Choice Test Item
If
find the value of csc A.
A
B
C
D
Read the Test Item
Draw a right triangle and label
one acute angle A. Since
and
, label the opposite
leg 5 and the adjacent leg 3.
Solve the Test Item
Use the Pythagorean Theorem to find c.
Pythagorean Theorem
Replace a with 3 and b with 5.
Simplify.
Take the square root
of each side.
Now find csc A.
Cosecant ratio
Replace hyp with
and opp with 5.
Answer: D
Multiple-Choice Test Item
If
find the value of cos B.
A
B
C
D
Answer: C
Write an equation involving sin, cos,
or tan that can be used to find the
value of x. Then solve the equation.
Round to the nearest tenth.
The measure of the hypotenuse is 12.
The side with the missing length is
opposite the angle measuring 60 . The
trigonometric function relating the
opposite side of a right triangle and the
hypotenuse is the sine function.
Sine ratio
Replace with 60 , opp with x,
and hyp with 12.
Multiply each side by 12.
Answer: The value of x is
or about 10.4.
Write an equation involving sin, cos,
or tan that can be used to find the
value of x. Then solve the equation.
Round to the nearest tenth.
Answer:
or about 8.7
Solving a General Right
Triangle
If you know the measures of any two sides
of a right triangle or the measures of one
side and one acute angle, you can
determine the measures of all the sides and
angles of the triangle. This process of
finding the missing measures is known as
solving a right triangle.
Solve XYZ. Round measures
of sides to the nearest tenth
and measures of angles to the
nearest degree.
x
z
You know the measures of one side, one acute angle,
and the right angle. You need to find x, z, and Y.
Find x and z.
Multiply each side by 11.
Use a calculator.
x
Multiply each side by 11.
Use a calculator.
z
Find Y.
Angles X and Y are
complementary.
Solve for Y.
Answer: Therefore,
,
, and
.
Solve XYZ. Round measures of sides
to the nearest tenth and measures of
angles to the nearest degree.
Answer:
Using the Inverse
• You can use the inverse capabilities to find
the measure of an angle when one of its
trigonometric ratios is known.
Solve ABC. Round measures
of sides to the nearest tenth
and measures of angles to the
nearest degree.
You know the measures of the sides. You need to
find A and B.
Find A.
Use a calculator and the SIN–1 function to find the angle
whose sine is
Keystrokes: 2nd [SIN–1] 9
17 )
ENTER
31.96571875
Find B.
Angles A and B are
complementary.
Solve for B.
Answer: Therefore,
and
.
Solve ABC. Round measures
of sides to the nearest tenth
and measures of angles to the
nearest degree.
Answer:
Uses
Trigonometry has many practical
applications. Among the most
important is the ability to find
distances or lengths that either
cannot be measured directly or are
not easily measured directly.
Bridge Construction In order to
construct a bridge across a river,
the width of the river must be
determined. A stake is planted on
one side of the river directly across
from a second stake on the
opposite side. At a distance 30
meters to the left of the stake, an
angle of 55 is measured between
the two stakes. Find the width of
the river.
Let w represent the width of the river. Write an
equation using a trigonometric function that involves
the ratio of w and 30.
Multiply each side by 30.
Use a calculator.
Answer: The width is about 42.8 meters.
Bridge Construction To construct a bridge, a stake
is planted on one side of the river directly across
from a second stake on the opposite side. The
angle between the two stakes is measured at a
distance 20 meters away from the stake, and found
to be 50 . Find the width of the river.
Answer: about 23.8 meters
Skiing A run has an angle of elevation of 15.7 and
a vertical drop of 1800 feet. Estimate the length of
this run.
Let represent the length of the run. Write an
equation using a trigonometric function that
involves the ratio of
and 1800.
Solve for
Use a calculator.
Answer: The length of the run is about 6652 feet.
Skiing A run has an angle of elevation of 23 and a
vertical drop of 1000 feet. Estimate the length of
this run.
Answer: about 2559 feet
Summary
• What does SOHCAHTOA stand for?
• What is the minimum information you have
to have about a right triangle to solve it?
• Assignment: finish packet pages 2-4 and
problems 29-40 on page 707 in the
textbook.
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