5.2.1

Unit Circle

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The Unit Circle

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

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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.
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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=cosθx=\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=sinθy=\sin \theta
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Sine and cosine

  • The tangent function is defined as the slope of the radius connecting the point and the origin:
    • tanθ=yx=sinθcosθ\tan\theta = \frac{y}{x} = \frac{sin\theta}{\cos\theta}
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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}.
    • sin90=sinπ2=1 \sin 90 = \sin \frac{\pi}{2}=1
    • cos90=cosπ2=0\cos 90 = \cos \frac{\pi}{2} = 0
    • tan90\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.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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