Chapter Contents

• Trigonometric Functions
• 1. Angles
• 2. Sine, Cosine, Tangent & Reciprocals
• 3. Values of Trigonometric Functions
• 4. The Right Triangle and Applications
• Trigonometry Revision Summary
• 5. Signs of the Trigonometric Functions
• 6. Trigonometric Functions of Any Angle
• 7. Radians
• 8. Applications of Radian Measure
• Pulleys
• 9. Radians and the Trigonometric Ratios
• Trigonometric Functions Problem Solver

• Comments, Questions?

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8. Applications of Radian Measure

by M. Bourne

In this section, we see some of the common applications of radian measure, including arc length, area of a sector of a circle, and angular velocity.

Go back to the section on Radians if you are not sure what is going on.

Arc Length

The length, s, of an arc of a circle radius r subtended by θ (in radians) is given by:

s = r θ

If r is in meters, s will also be in meters. Likewise, if r is in cm, s will also be in cm.

Example 1

Find the length of the arc of a circle with radius `4\ "cm"` and central angle `5.1` radians.

Answer

Area of a Sector

The area of a sector with central angle θ (in radians) is given by:

`"Area"=(theta\ r^2)/2`

If r is in `"m"`, the area will be measured in `"m"`². If r is in `"cm"`, area will be in `"cm"`².

Example 2

Find the area of the sector with radius `7\ "cm"` and central angle `2.5` radians.

Answer

Angular Velocity

The time rate of change of angle θ by a rotating body is the angular velocity, written ω (omega). It is measured in radians/second.

If v is the linear velocity (in m/s) and r is the radius of the circle (in m), then

v = rω

Note: If r is in `"cm"`, v will be in `"cm/s"`.

Example 3

A bicycle with tyres `90\ "cm"` in diameter is travelling at `25` km/h. What is the angular velocity of the tyre in radians per second?

Answer

Flash Game

The man runs at a constant velocity and a ball is revolving overhead.

You increase points if:

• the linear velocity of the ball is more than 6 m/sec - add 1 point
• the ball is close to the man (within one body length) - add 10 points

Your points go down if:

• the ball goes outside the borders - minus 1 point
• the ball hurts the man - minus 20 points

You can use

• the first slider or the right and left arrows on your keyboard to change the radius and
• the second slider or the up and down arrows to change the angular velocity.

Exercises:

1. A section of side walk is a circular sector of radius `1.25\ "m"` and central angle `50.6°`. What is the area of this section of sidewalk?

Answer

2. A cam is in the shape of a circular sector with radius `1.875\ "cm"` and central angle `165.58°`. What is the perimeter of the cam?

Answer

3. The roller on a computer printer makes `2200` rev/min. What is its angular velocity?

Answer

4. The propeller on a motorboat is rotating at `130` rad/s. What is the linear velocity of a point on the tip of a blade if the blade is `22.5` cm long?

Answer

5. The sweep second hand of a watch is `15.0` mm long. What is the linear velocity of the tip?

Answer

Pulley Problems

You can investigate the linear velocity of a belt moving around two pulleys in this Flash example.

Go to Pulleys simulation.