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Unit Circle

Maths: Edexcel A Level Pure Maths, topic 5.2.1

The Unit Circle

A unit circle is a circle centered at the origin with a radius of 1.

Radians

Radians

  • An angle of one radian is created at the center of a unit circle by two radii and an arclength of 1.
  • The arclength of a unit circle is then equal to the angle θ\theta.
  • s=rθ=θs = r\theta=\theta
  • This means we can write a general point on the circle (x,y)(x,y) as a function of θ\theta.
Trigonometric functions

Trigonometric functions

  • The cosine function is defined as the function that takes the angle from the positive xx-axis as its input and gives the xx-coordinate of a point on the unit circle:
  • x=cos⁡θx=\cos \theta
  • Likewise, the sine function is defined as the function that takes the same angle and gives the corresponding yy-coordinate of the point on the unit circle.
  • y=sin⁡θy=\sin \theta
Sine and cosine

Sine and cosine

  • The tangent function is defined as the slope of the radius connecting the point and the origin:
  • tan⁡θ=yx=sinθcos⁡θ\tan\theta = \frac{y}{x} = \frac{sin\theta}{\cos\theta}
Special coordinates

Special coordinates

  • The point on the unit circle with coordinates (0,1)(0,1) corresponds to an angle of 90° or π2\frac{\pi}{2}.
  • sin⁡90=sin⁡π2=1 \sin 90 = \sin \frac{\pi}{2}=1
  • cos⁡90=cos⁡π2=0\cos 90 = \cos \frac{\pi}{2} = 0
  • tan⁡90\tan 90 is undefined.
  • By continuing around the circle, we can define the sine and cosine of all real values of θ\theta this way as we go around the circle.

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