13-3 Radian Measure (p. 726)

Algebra 2Prentice Hall, 2007

Objectives… You will:

ContentRecall geometric terms related to the

parts of a circle and use them to convert degrees to radians and vice versa.

Use the circumference formula to determine arc length.

LanguageRecognize when an angle’s measure is

written in degrees or radians.

Recall from Geometry…

A central angle of a circle is an angle with its vertex at the center of a circle.

An intercepted arc is the portion of the circle with endpoints on the sides of the central angle.

Special Angle Measurements…

When a central angle intercepts an arc that has the same length as the radius of the circle, the measure of the angle is defined to be 1 radian.

Special Angle Measurements…

Because the circumference of a circle is 2r (and in a unit circle 1 radius = 1 radian), there are 2 radians in every circle.

360 degrees = 2 radians

Think…

What do you see?

Conversion Factors…Remember “unit analysis”… where you

multiply by whatever fraction it takes to get rid of the old unit and turn it into the new one?

✕ new unit /old unit --- cancel 1st top with 2nd bottom

Ex. Convert 32 feet to inches.

Conversion Factors…Degrees to radians multiply by π

radians/180°

Radians to degrees multiply by 180°/π

radians

Examples…1. Find the radian measure for each angle:

45 180 330

2. Find the degree measure for each angle: /3 radians 3/4 radians -2/3 radians

3. NOW, find the radian measure for all the angles on your Unit Circle! HINT: start with the axes…

Arc Length (with Degrees)…

€

∠360°

×2πr

In Geometry, you found the length of an intercepted arc by multiplying the circumference by the fraction of the circle:

Arc Length (and Radians)…

When your angle is given in radians, simply multiply the radius by the radian measure:

radians radius

Your Turn…

Ex. 4 Find the lengths of s and b:

Real World Example

Ex. 5 A weather satellite in a circular orbit around Earth completes one orbit every 2 hours. The radius of Earth is about 6400 km, and the satellite orbits 2600 km above Earths’ surface. How far does the satellite travel in 1 hour?

Assignment… 13-3 p. 729: mult of 3 (3-42,

not 27);47, 48, 74