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9-5. Trigonometric Ratios. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. 9.5 Trigonometric Ratios. Warm Up Write 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. - PowerPoint PPT PresentationTRANSCRIPT
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
9.5 Trigonometric Ratios
trigonometric ratiosinecosineTangentAngle of ElevationAngle of Depression
Vocabulary
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
9.5 Trigonometric Ratios
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
9.5 Trigonometric Ratios
Example 1A: Finding Trigonometric Ratios
Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.
sin J
9.5 Trigonometric Ratios
cos J
Example 1B: Finding Trigonometric Ratios
Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.
9.5 Trigonometric Ratios
tan K
Example 1C: Finding Trigonometric Ratios
Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.
9.5 Trigonometric Ratios
Check It Out! Example 1a
Write the trigonometric ratio as a fraction and as a decimal rounded tothe nearest hundredth.
cos A
9.5 Trigonometric Ratios
Check It Out! Example 1b
Write the trigonometric ratio as a fraction and as a decimal rounded tothe nearest hundredth.
tan B
9.5 Trigonometric Ratios
Check It Out! Example 1c
Write the trigonometric ratio as a fraction and as a decimal rounded tothe nearest hundredth.
sin B
9.5 Trigonometric Ratios
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 ∆.
9.5 Trigonometric Ratios
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
9.5 Trigonometric Ratios
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!
9.5 Trigonometric Ratios
Example 3B: Calculating Trigonometric Ratios
Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.
cos 19°
cos 19° 0.95
9.5 Trigonometric Ratios
Example 3C: Calculating Trigonometric Ratios
Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.
tan 65°
tan 65° 2.14
9.5 Trigonometric Ratios
Check It Out! Example 3a
Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.
tan 11°
tan 11° 0.19
9.5 Trigonometric Ratios
Check It Out! Example 3b
Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.
sin 62°
sin 62° 0.88
9.5 Trigonometric Ratios
Check It Out! Example 3c
Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.
cos 30°
cos 30° 0.87
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator.
Caution!
9.5 Trigonometric Ratios
Example 4B: Using Trigonometric Ratios to Find Lengths
Find the length. Round to the 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.
9.5 Trigonometric Ratios
Example 4B Continued
Write a trigonometric ratio.
Substitute the given values.
12.9(sin 63°) = QR
11.49 cm QR
Multiply both sides by 12.9.
Simplify the expression.
9.5 Trigonometric Ratios
Example 4C: Using Trigonometric Ratios to Find Lengths
Find the length. Round to the 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.
9.5 Trigonometric Ratios
Example 4C Continued
Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by FD and divide by cos 39°.
Simplify the expression.FD 25.74 m
9.5 Trigonometric Ratios
Check It Out! Example 4a
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.
9.5 Trigonometric Ratios
Check It Out! Example 4a Continued
Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by DF and divide by sin 51°.
Simplify the expression.DF 21.87 cm
9.5 Trigonometric Ratios
Check It Out! Example 4b
Find the length. Round to the nearest 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.
9.5 Trigonometric Ratios
Check It Out! Example 4b Continued
Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by 9.5.
Simplify the expression.
ST = 9.5(cos 42°)
ST 7.06 in.
9.5 Trigonometric Ratios
Check It Out! Example 4c
Find the length. Round to the 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.
9.5 Trigonometric Ratios
Check It Out! Example 4c Continued
Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by BC and divide by tan 18°.
Simplify the expression.BC 36.93 ft
9.5 Trigonometric Ratios
Check It Out! Example 4d
Find the length. Round to the nearest 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.
9.5 Trigonometric Ratios
Check It Out! Example 4d Continued
Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by 13.6.
Simplify the expression.
JL = 13.6(sin 27°)
JL 6.17 cm
9.5 Trigonometric Ratios
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?
9.5 Trigonometric Ratios
11 Understand the Problem
Make a sketch. The answer is BC.
Example 5 Continued
0.9 km
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
Solve33
Example 5 Continued
Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by CA and divide by sin 25.6°.
Simplify the expression.CA 2.0829 km
9.5 Trigonometric Ratios
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
9.5 Trigonometric Ratios
Check It Out! Example 5
Find AC, the length of the ramp, to the nearest hundredth of a foot.
9.5 Trigonometric Ratios
Check It Out! Example 5 Continued
11 Understand the Problem
Make a sketch. The answer is AC.
9.5 Trigonometric Ratios
22 Make a Plan
Check It Out! Example 5 Continued
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.
9.5 Trigonometric Ratios
Solve33
Check It Out! Example 5 Continued
Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by AC and divide by sin 4.8°.
Simplify the expression.AC 14.3407 ft
9.5 Trigonometric Ratios
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.
Check It Out! Example 5 Continued
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
Check It Out! Example 1
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
Check It Out! Example 2
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
Check It Out! Example 3
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°
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
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.
9.5 Trigonometric Ratios
Example 4 Continued
Step 2 Find y.
By the Alternate Interior Angles Theorem, mCAL = 58°.
.
In ∆ALC,
So
9.5 Trigonometric Ratios
Step 3 Find z.
By the Alternate Interior Angles Theorem, mCBL = 22°.
Example 4 Continued
In ∆BLC,
So
9.5 Trigonometric Ratios
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
9.5 Trigonometric Ratios
Check It Out! Example 4
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.
9.5 Trigonometric Ratios
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.
Check It Out! Example 4 Continued
78°19°
78° 19°
12,000 ft
9.5 Trigonometric Ratios
Step 2 Find y.
By the Alternate Interior Angles Theorem, mCAP = 78°.
Check It Out! Example 4 Continued
In ∆APC,
So
9.5 Trigonometric Ratios
Step 3 Find z.
By the Alternate Interior Angles Theorem, mCBP = 19°.
Check It Out! Example 4 Continued
In ∆BPC,
So
9.5 Trigonometric Ratios
Step 4 Find x.
So the two airports are about 32,300 ft apart.
Check It Out! Example 4 Continued
x = z – y
x 34,851 – 2551 32,300 ft
9.5 Trigonometric Ratios
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
9.5 Trigonometric Ratios
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
9.5 Trigonometric Ratios
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
9.5 Trigonometric Ratios
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
9.5 Trigonometric Ratios