13.3 – Radian Measures. Radian Measure Find the circumference of a circle with the given radius or...
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Transcript of 13.3 – Radian Measures. Radian Measure Find the circumference of a circle with the given radius or...
![Page 1: 13.3 – Radian Measures. Radian Measure Find the circumference of a circle with the given radius or diameter. Round your answer to the nearest tenth. 1.radius.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649da25503460f94a8e642/html5/thumbnails/1.jpg)
13.3 – Radian Measures
![Page 2: 13.3 – Radian Measures. Radian Measure Find the circumference of a circle with the given radius or diameter. Round your answer to the nearest tenth. 1.radius.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649da25503460f94a8e642/html5/thumbnails/2.jpg)
Radian MeasureFind the circumference of a circle with the given radius or diameter. Round your answer to the nearest tenth.
1. radius 4 in. 2. diameter 70 m
3. radius 8 mi 4. diameter 3.4 ft
5. radius 5 mm 6. diameter 6.3 cm
![Page 3: 13.3 – Radian Measures. Radian Measure Find the circumference of a circle with the given radius or diameter. Round your answer to the nearest tenth. 1.radius.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649da25503460f94a8e642/html5/thumbnails/3.jpg)
Radian Measure1. C = 2 r = 2 (4 in.) 25.1 in.
2. C = d = (70 m) 219.9 m
3. C = 2 r = 2 (8 mi) 50.3 mi
4. C = d = (3.4 ft) 10.7 ft
5. C = 2 r = 2 (5 mm) 31.4 mm
6. C = d = (6.3 cm) 19.8 cm
Solutions
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Vocabulary and Definitions
• A central angle of a circle is an angle with a vertex at the center of the circle.
• An intercepted arc is the arc that is “captured” by the central angle.
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Vocabulary and Definitions
• When the central angle intercepts an arc that has the same length as a radius of the circle, the measure of the angle is defined as a radian.
r
r
Like degrees, radians measure the amount of rotation from the initial side to the terminal side of the angle.
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The Unit CircleThe Unit Circle
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The “Magic” ProportionThis proportion can be used to convert to and from Degrees to Radians.
Degrees°180° =
r radians radians
Example: Find the radian measure of angle of 45°.
Write a proportion.45°180° =
r radians radians
An angle of 45° measures about 0.785 radians.
Write the cross-products.45 • = 180 • r
Divide each side by 45.r = 45 •180
= 0.785 Simplify.4
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The “Magic” ProportionThis proportion can be used to convert to and from Degrees to Radians.
Degrees°180° =
r radians radians
Example: Find the radian measure of angle of -270°.
Write a proportion.-270°180° =
r radians radians
An angle of -270° measures about -4.71 radians.
Write the cross-products.-270 • = 180 • r
Divide each side by 45.r = -270 •180
-4.71 Simplify.2-3
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Let’s Try SomeConvert the following to radiansa. 390o b. 54o c. 180o
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Example
= 390° Simplify.
Find the degree measure of .6
13
Write a proportion.613
radians= d°
180
• 180 = • d Write the cross-product.6
13
d = Divide each side by .13 • 180
6 •1
30
An angle of radians measures 390°.6
13
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Example Find the degree measure of an angle of – radians.
23
= –270°
An angle of – radians measures –270°.2
3
– radians • = – radians •2
3 180°radians 2
3 180°radians1
90Multiply by
180°radians .
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Radian Measure
Find the radian measure of an angle of 54°.
5 4° • radians = 54° • radians Multiply by radians.180° 180° 180°3
10
103 radians= Simplify.
An angle of 54° measures radians.103
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Draw the angle.
Radian MeasureFind the exact values of cos and sin .radians3 radians3
radians • = 60° Convert to degrees.3180°radians
Complete a 30°-60°-90° triangle.
The hypotenuse has length 1.
radians3Thus, cos =12
and sin radians3 = . 32
The shorter leg is the length of the hypotenuse, and the longer leg is 3 times the length of the shorter leg.
12
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Radian MeasureUse this circle to find length s to the nearest tenth.
s = r Use the formula.
The arc has length 22.0 in.
= 7 Simplify.
22.0 Use a calculator.
= 6 • Substitute 6 for r and for .7 6
7 6
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Radian MeasureAnother satellite completes one orbit around Earth every 4 h. The
satellite orbits 2900 km above Earth’s surface. How far does the
satellite travel in 1 h?
Since one complete rotation (orbit) takes 4 h, the satellite completes of a rotation in 1 h.
14
Step 1: Find the radius of the satellite’s orbit.
r = 6400 + 2900 Add the radius of Earth and the distance
from Earth’s surface to the satellite.
= 9300
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Radian Measure(continued)
The satellite travels about 14,608 km in 1 h.
Step 2: Find the measure of the central angle the satellite travels through in 1 h.
= • 2 Multiply the fraction of the rotation by the number of radians in one complete rotation.
= • Simplify.
14
12
Step 3: Find s for = .
s = r Use the formula.
= 9300 • Substitute 9300 for r and for .
14608 Simplify.
2
2 2