1. Lesson 1

1.5. Explore

Mathematics 30-2 Module 6

Module 6: Sinusoidal Functions

 

Explore

 

This is a sketch of part of a circle. The centre is marked as O and a radius is drawn to point A on the circle. Another radius is drawn to point B on the circle. Points A and B are separated by an arc length equal to one radius. The central angle created at AOB is labelled as 1 radian. A label indicates that arc length is equal to radius.

In Try This 1 you created a central angle similar to the one in the diagram shown here. Here, the radius of the circle, AO, is equal to the length of the subtended arc, AB. Remember that an arc can be defined as a part of the circumference of a circle. This situation defines radian measure. The measure of ∠AOB is 1 rad (radian) since the measure of arc AB is equal to the radius of the circle.

 

When an angle measurement is given and there are no units written after the measurement, you can assume the units are in radians. When you’re writing an angle measurement in degrees, the degree symbol must be included to indicate the measurement is in degrees. For example,

  • 75° indicates 75 degrees
  • 110 indicates 110 radians (rad)

In Try This 1 you may have determined that the radius of the circle would fit about six times around its circumference. This would mean that one full rotation, which is 360°, would be approximately the same as 6 rad.

 

After manipulating the formula C = 2πr, you get When you divide the circumference by the radius measure, you can conclude that 2π, or 6.283…, is the radian measure of a circle. Since you know that there are 360° in a circle, this means 2π is equal to 360°. However, for most of this lesson, the approximation 360° = 6 rad or 60° = 1 rad will be used.

 

In Try This 2 you will use this information to estimate the size of an angle in radians given an angle measured in degrees.

 

The first diagram shows the size of angles with measures 0°, 90°, 180°, 270°, and 360°. The second diagram shows the size of angles with measures 0, 1, 2, 3, π, 4, 5, 6 and 2π radians.

 

Try This 2
  1. Draw a sketch of each of the following angles. Estimate the size of each angle in radians.
    1. 120°
    2. 210°
    3. 341°
  1.  

    1. Open Angles in Degrees and Radians. Move the angle slider to check your estimates for question 1. The left and right arrow keys can be used to change the angle by single degrees. The angles you drew may be of different orientations to those in the program, but they should be a similar size.

       

       

      This is a play button that opens Angles in Degrees and Radians.

    2. Does the approximation that 1 rad is about 60° produce good estimates?

course folder Save your responses in your course folder.

Use either the fact that 360° = 2π rad or that 60° is approximately 1 rad.