Trig ratios

8-2 Trigonometric Ratios

Warm Up
Lesson Presentation
Lesson Quiz

Holt Geometry
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Warm Up
Write each fraction as a decimal rounded to
the nearest hundredth.
1.

2.

0.67

0.29

Solve each equation.
3.

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x = 7.25

4.

x = 7.99


Objectives

Find the sine, cosine, and tangent of an acute
angle.
Use trigonometric ratios to find side lengths in
right triangles and to solve real-world
problems.

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Vocabulary

trigonometric ratio
sine
cosine
tangent

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By the AA Similarity Postulate, a right triangle with
a given acute angle is similar to every other right triangle with
that same acute angle measure. So ∆ABC ~ ∆DEF ~ ∆XYZ,
and
. These are trigonometric ratios. A
trigonometric ratio is a ratio of two sides of a right triangle.

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Writing Math
In trigonometry, the letter of the vertex of the angle is
often used to represent the measure of that angle. For
example, the sine of ∠A is written as sin A.

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Example 1A: Finding Trigonometric Ratios
Write the trigonometric
ratio as a fraction and as a
decimal rounded to the
nearest hundredth.
sin J

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Example 1B: Finding Trigonometric Ratios
nearest hundredth.
cos J

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Example 1C: Finding Trigonometric Ratios
nearest hundredth.
tan K

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Check It Out! Example 1a
decimal rounded to
cos A

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Check It Out! Example 1b
decimal rounded to
tan B

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Check It Out! Example 1c
decimal rounded to
sin B

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Example 2: Finding Trigonometric Ratios in Special
Right Triangles
Use a special right triangle to write cos 30° as a
fraction.
Draw and label a 30º-60º-90º ∆.

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Check It Out! Example 2
Use a special right triangle to write tan 45° as a
fraction.
Draw and label a 45º-45º-90º ∆.

45°
s
45°
s

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Example 3A: Calculating Trigonometric Ratios
Use your calculator to find the trigonometric ratio.
Round to the nearest hundredth.
sin 52°
Caution!
Be sure your
calculator is in
degree mode, not
radian mode.
sin 52° ≈ 0.79
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Example 3B: Calculating Trigonometric Ratios
cos 19°

cos 19° ≈ 0.95
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Example 3C: Calculating Trigonometric Ratios
tan 65°

tan 65° ≈ 2.14
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tan 11°

tan 11° ≈ 0.19
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sin 62°

sin 62° ≈ 0.88
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cos 30°

cos 30° ≈ 0.87

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The hypotenuse is always the longest side of a right
triangle. So the denominator of a sine or cosine ratio is
always greater than the numerator. Therefore the sine
and cosine of an acute angle are always positive
numbers less than 1. Since the tangent of an acute
angle is the ratio of the lengths of the legs, it can have
any value greater than 0.

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Example 4A: Using Trigonometric Ratios to Find
Lengths
Find the length. Round to the
nearest hundredth.
BC
is adjacent to the given angle, ∠B. You are given AC,
which is opposite ∠B. Since the adjacent and opposite legs
are involved, use a tangent ratio.

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Example 4A Continued

Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by BC and
divide by tan 15°.
BC ≈ 38.07 ft
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Simplify the expression.


Caution!
Do not round until the final step of your answer. Use the
values of the trigonometric ratios provided by your
calculator.

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Example 4B: Using Trigonometric Ratios to Find
Lengths
nearest hundredth.
QR
is opposite to the given angle, ∠P. You are given PR,
which is the hypotenuse. Since the opposite side and
hypotenuse are involved, use a sine ratio.

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Example 4B Continued

12.9(sin 63°) = QR
11.49 cm ≈ QR
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Multiply both sides by 12.9.

Example 4C: Using Trigonometric Ratios to Find
Lengths
nearest hundredth.
FD
is the hypotenuse. You are given EF, which is adjacent to
the given angle, ∠F. Since the adjacent side and hypotenuse
are involved, use a cosine ratio.

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Example 4C Continued

Multiply both sides by FD and divide
by cos 39°.
FD ≈ 25.74 m
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Find the length. Round to the nearest
hundredth.
DF

is the hypotenuse. You are given EF, which is opposite to
the given angle, ∠D. Since the opposite side and hypotenuse
are involved, use a sine ratio.

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Check It Out! Example 4a Continued


DF ≈ 21.87 cm
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Multiply both sides by DF and divide
by sin 51°.

hundredth.
ST

is a leg. You are given TU, which is the hypotenuse. Since
the adjacent side and hypotenuse are involved, use a cosine
ratio.

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Check It Out! Example 4b Continued

ST = 9.5(cos 42°)
ST ≈ 7.06 in.
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nearest hundredth.
BC

is a leg. You are given AC, which is the opposite side to
given angle, ∠B. Since the opposite side and adjacent side
are involved, use a tangent ratio.

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Check It Out! Example 4c Continued

Multiply both sides by BC and divide
by tan 18°.
BC ≈ 36.93 ft
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Check It Out! Example 4d
hundredth.
JL

is the opposite side to the given angle, ∠K. You are given
KL, which is the hypotenuse. Since the opposite side and
hypotenuse are involved, use a sine ratio.

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Check It Out! Example 4d Continued

JL = 13.6(sin 27°)
JL ≈ 6.17 cm
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Example 5: Problem-Solving Application

The Pilatusbahn in Switzerland is the world’s
steepest cog railway. Its steepest section makes an
angle of about 25.6º with the horizontal and rises
about 0.9 km. To the nearest hundredth of a
kilometer, how long is this section of the railway
track?

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Example 5 Continued
1

Understand the Problem
Make a sketch. The answer is BC.

0.9 km

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Example 5 Continued
2

Make a Plan

is the hypotenuse. You are given BC, which is the leg
opposite ∠A. Since the opposite and hypotenuse are
involved, write an equation using the sine ratio.

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Example 5 Continued
3

Solve

Multiply both sides by CA and divide
by sin 25.6°.
CA ≈ 2.0829 km
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Example 5 Continued
4

Look Back
The problem asks for CA rounded to the nearest
hundredth, so round the length to 2.08. The section
of track is 2.08 km.

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Check It Out! Example 5
Find AC, the length of the ramp, to the nearest
hundredth of a foot.

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Check It Out! Example 5 Continued
1

Understand the Problem
Make a sketch. The answer is AC.

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2

Make a Plan
is the hypotenuse to ∠C. You are given AB,
which is the leg opposite ∠C. Since the opposite
leg and hypotenuse are involved, write an
equation using the sine ratio.

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3

Solve

Multiply both sides by AC and divide
by sin 4.8°.
AC ≈ 14.3407 ft

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4

Look Back
The problem asks for AC rounded to the nearest
hundredth, so round the length to 14.34. The length
of ramp covers a distance of 14.34 ft.

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Lesson Quiz: Part I
Use a special right triangle to write each
trigonometric ratio as a fraction.
1. sin 60°

2. cos 45°

Use your calculator to find each trigonometric
ratio. Round to the nearest hundredth.
3. tan 84°

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9.51

4. cos 13°

0.97

Lesson Quiz: Part II
Find each length. Round to the nearest tenth.
5. CB

6.1

6. AC

16.2

Use your answers from Items 5 and 6 to write
each trigonometric ratio as a fraction and as a
decimal rounded to the nearest hundredth.
7. sin A
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8. cos A

9. tan A

Trig ratios

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