9-5

9-5 Trigonometric Ratios

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Warm UpWrite each fraction as a decimal rounded to the nearest hundredth.

1. 2.

Solve each equation.

3. 4.

0.67 0.29

x = 7.25 x = 7.99

9.5 Trigonometric Ratios

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.

Solve problems involving angles of elevation and angles of depression

Objectives


trigonometric ratiosinecosineTangentAngle of ElevationAngle of Depression

Vocabulary


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.



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.

Writing Math


Example 1A: Finding Trigonometric Ratios

Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

sin J


cos J

Example 1B: Finding Trigonometric Ratios



tan K

Example 1C: Finding Trigonometric Ratios



Check It Out! Example 1a

Write the trigonometric ratio as a fraction and as a decimal rounded tothe nearest hundredth.

cos A


Check It Out! Example 1b


tan B


Check It Out! Example 1c


sin B


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 ∆.


Check It Out! Example 2

Use a special right triangle to write tan 45° as a fraction.

Draw and label a 45º-45º-90º ∆.

s

45°

45°

s


Example 3A: Calculating Trigonometric Ratios

Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.

sin 52°

sin 52° 0.79

Be sure your calculator is in degree mode, not radian mode.

Caution!


Example 3B: Calculating Trigonometric Ratios


cos 19°

cos 19° 0.95


Example 3C: Calculating Trigonometric Ratios


tan 65°

tan 65° 2.14




tan 11°

tan 11° 0.19




sin 62°

sin 62° 0.88




cos 30°

cos 30° 0.87


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.


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.


Example 4A Continued

BC 38.07 ft

Write a trigonometric ratio.

Substitute the given values.

Multiply both sides by BC and divide by tan 15°.

Simplify the expression.


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

Caution!


Example 4B: Using Trigonometric Ratios to Find Lengths


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.


Example 4B Continued



12.9(sin 63°) = QR

11.49 cm QR

Multiply both sides by 12.9.



Example 4C: Using Trigonometric Ratios to Find Lengths


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.


Example 4C Continued



Multiply both sides by FD and divide by cos 39°.

Simplify the expression.FD 25.74 m




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.


Check It Out! Example 4a Continued



Multiply both sides by DF and divide by sin 51°.

Simplify the expression.DF 21.87 cm




ST

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


Check It Out! Example 4b Continued





ST = 9.5(cos 42°)

ST 7.06 in.




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.


Check It Out! Example 4c Continued



Multiply both sides by BC and divide by tan 18°.

Simplify the expression.BC 36.93 ft


Check It Out! Example 4d


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.


Check It Out! Example 4d Continued





JL = 13.6(sin 27°)

JL 6.17 cm


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?


11 Understand the Problem

Make a sketch. The answer is BC.

Example 5 Continued

0.9 km


22 Make a Plan

Example 5 Continued

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.


Solve33

Example 5 Continued



Multiply both sides by CA and divide by sin 25.6°.

Simplify the expression.CA 2.0829 km


Look Back44

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.

Example 5 Continued



Find AC, the length of the ramp, to the nearest hundredth of a foot.


Check It Out! Example 5 Continued

11 Understand the Problem

Make a sketch. The answer is AC.


22 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.


Solve33




Multiply both sides by AC and divide by sin 4.8°.

Simplify the expression.AC 14.3407 ft


Look Back44

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.



Since horizontal lines are parallel, 1 2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruentto the angle of depression from the other point.


Angles of Elevation and Depression:

Example 1A: Classifying Angles of Elevation and Depression

Classify each angle as an angle of elevation or an angle of depression.

1

1 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression.


Example 1B: Classifying Angles of Elevation and Depression

Classify each angle as an angle of elevation or an angle of depression.

4

4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.



Use the diagram above to classify each angle as an angle of elevation or angle of depression.

1a. 5

1b. 6

6 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

5 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression.


Example 2: Finding Distance by Using Angle of Elevation

The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 70º, how tall is the Space Needle? Round to the nearest meter.

Draw a sketch to represent the given information. Let A represent the tip of the shadow, and let B represent the top of the Space Needle. Let y be the height of the Space Needle.


Example 2 Continued

You are given the side adjacent to A, and y is the side opposite A. So write a tangent ratio.

y = 67 tan 70° Multiply both sides by 67.

y 184 m Simplify the expression.



What if…? Suppose the plane is at an altitude of 3500 ft and the angle of elevation from the airport to the plane is 29°. What is the horizontal distance between the plane and the airport? Round to the nearest foot.

3500 ft

29°

You are given the side opposite A, and x is the side adjacent to A. So write a tangent ratio.

Multiply both sides by x and divide by tan 29°.

x 6314 ft Simplify the expression.


Example 3: Finding Distance by Using Angle of Depression

An ice climber stands at the edge of a crevasse that is 115 ft wide. The angle of depression from the edge where she stands to the bottom of the opposite side is 52º. How deep is the crevasse at this point? Round to the nearest foot.


Example 3 Continued

Draw a sketch to represent the given information. Let C represent the ice climber and let B represent the bottom of the opposite side of the crevasse. Let y be the depth of the crevasse.


Example 3 Continued

By the Alternate Interior Angles Theorem, mB = 52°.

Write a tangent ratio.

y = 115 tan 52° Multiply both sides by 115.

y 147 ft Simplify the expression.



What if…? Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot.

By the Alternate Interior Angles Theorem, mF = 3°.

Write a tangent ratio.

Multiply both sides by x and divide by tan 3°.

x 1717 ft Simplify the expression.

3°


Example 4: Shipping Application

An observer in a lighthouse is 69 ft above the water. He sights two boats in the water directly in front of him. The angle of depression to the nearest boat is 48º. The angle of depression to the other boat is 22º. What is the distance between the two boats? Round to the nearest foot.


Example 4 Application

Step 1 Draw a sketch. Let L represent the observer in the lighthouse and let A and B represent the two boats. Let x be the distance between the two boats.


Example 4 Continued

Step 2 Find y.

By the Alternate Interior Angles Theorem, mCAL = 58°.

.

In ∆ALC,

So


Step 3 Find z.

By the Alternate Interior Angles Theorem, mCBL = 22°.

Example 4 Continued

In ∆BLC,

So


Step 4 Find x.

So the two boats are about 109 ft apart.

Example 4 Continued

x = z – y

x 170.8 – 62.1 109 ft



A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot.


Step 1 Draw a sketch. Let P represent the pilot and let A and B represent the two airports. Let x be the distance between the two airports.


78°19°

78° 19°

12,000 ft


Step 2 Find y.

By the Alternate Interior Angles Theorem, mCAP = 78°.


In ∆APC,

So


Step 3 Find z.

By the Alternate Interior Angles Theorem, mCBP = 19°.


In ∆BPC,

So


Step 4 Find x.

So the two airports are about 32,300 ft apart.


x = z – y

x 34,851 – 2551 32,300 ft


Lesson Quiz: Part I

Classify each angle as an angle of elevation or angle of depression.

1. 6

2. 9

angle of depression

angle of elevation


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° 4. cos 13° 9.51 0.97


Lesson Quiz: Part II

Find each length. Round to the nearest tenth.

5. CB

6. AC

6.1

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 8. cos A 9. tan A


Lesson Quiz: Part II

3. A plane is flying at an altitude of 14,500 ft. The angle of depression from the plane to a control tower is 15°. What is the horizontal distance from the plane to the tower? Round to the nearest foot.

4. A woman is standing 12 ft from a sculpture. The angle of elevation from her eye to the top of the sculpture is 30°, and the angle of depression to its base is 22°. How tall is the sculpture to the nearest foot?

54,115 ft

12 ft


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