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Degrees and Radians

Measurement of Angles

On a Cartesian coordinate system, we draw a circle centred at the origin and with radius 1.

Let P be a point on the circle.

The line OP and the positive x-axis form an angle

Let us agree that an angle measured counter-clockwise is positive, and an angle measured clockwise is negative. (See diagram below.)



Starting from the positive x-axis, the point P travels around the circle in a counter-clockwise manner.

When it arrives at the positive y-axis (a quarter of the way around the circle) its angle measures 90°.

When it reaches the negative x-axis (and is now halfway around the circle), its angle measures 180°.

When it hits the negative y-axis (and is three quarters of the way around the circle) it angle measures 270°.

And when it returns to the positive x-axis (and has completed a full circle) its angle measures 360°.

If the point travels clockwise, when it hits the negative y-axis its angle will measure -90° and when it hits the negative x-axis its angle will measure -180°.



Exercise 1. 

Enter the number of degrees of the angle in each case:

Press Tab key to check your answer.

P = (x,y)

Angle Size

Angle Size

x = 1, y = 0
x > 0, x = y
x = 0, y = 1
x < 0, y = -x
x = -1, y = 0
x < 0, y = x
x = 0, y = -1
x > 0, y = -x



If the length of the arc from (1, 0) to P is 1unit we say the angle is 1 radian ( rad in short). When the unit of an angle is absent, we assume it is in radians.

The circumference of the circle of radius 1 is 2π or 2π radians.

So, measuring counter-clockwise,

2π = 360° and π = 180°.

This provides us with the following conversion formulas for changing between degrees D and radians R:



Exercise 2.


Fill in the blanks (type pi for π) to convert each of the following:


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