More Basic Functions and Graphs: Background You’ll Need 1

  • Evaluate trigonometric functions using the unit circle

Trigonometric Functions and the Unit Circle

When evaluating trigonometric functions, the unit circle is an invaluable tool. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to a right triangle, where the hypotenuse is the radius of the circle, and the [latex]x[/latex] and [latex]y[/latex] coordinates of the point represent the lengths of the other two sides.

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Figure 1. Unit Circle

The unit circle provides the sine and cosine values for any given angle measure. For each angle, the [latex]x[/latex]-coordinate represents its cosine value, and the [latex]y[/latex]-coordinate stands for its sine value. The tangent of an angle is the ratio of the sine to the cosine:

[latex]\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{y}{x}[/latex]

unit circle

The unit circle allows us to evaluate trigonometric functions by using the coordinates of points on the circle.

  • The [latex]x[/latex]-coordinate gives the cosine value
  • The [latex]y[/latex]-coordinate gives the sine value
  • The ratio of [latex]y[/latex] to [latex]x[/latex] gives the tangent value for any given angle.

Remember, every angle in quadrant two, three, or four has a reference angle that lies in quadrant one. The quadrant of the original angle only affects the sign (positive or negative) of a trigonometric function’s value at a given angle.

Certain angles have coordinates that can be easily remembered:

  • At [latex]0[/latex] degrees (or [latex]0[/latex] radians), the coordinates are ([latex]1, 0[/latex]), so [latex]\cos(0) = 1[/latex] and [latex]\sin(0) = 0[/latex].
  • At [latex]90[/latex] degrees (or [latex]\frac{\pi}{2}[/latex] radians), the coordinates are ([latex]0, 1[/latex]), so [latex]\cos\left(\frac{\pi}{2}\right) = 0[/latex] and [latex]\sin\left(\frac{\pi}{2}\right) = 1[/latex].
  • At [latex]180[/latex] degrees (or [latex]\pi[/latex] radians), the coordinates are ([latex]-1, 0[/latex]), so [latex]\cos(\pi) = -1[/latex] and [latex]\sin(\pi) = 0[/latex].
  • At [latex]270[/latex] degrees (or [latex]\frac{3\pi}{2}[/latex] radians), the coordinates are ([latex]0, -1[/latex]), so [latex]\cos\left(\frac{3\pi}{2}\right) = 0[/latex] and [latex]\sin\left(\frac{3\pi}{2}\right) = -1[/latex].

How To: Evaluate at Any Angle using the Unit Circle

For any angle [latex]\theta[/latex], you can determine its corresponding point on the unit circle by:

  1. Start with your given angle [latex]\theta[/latex]. Position it so that it starts at the positive [latex]x[/latex]-axis and opens counterclockwise for positive angles, or clockwise for negative angles.
  2. Extend the angle’s terminal side until it intersects the unit circle at a point [latex]P[/latex].
  3. The coordinates of point [latex]P(x,y)[/latex] on the unit circle give you the cosine and sine of [latex]\theta[/latex] respectively.
    • [latex]\cos(\theta)[/latex] is the [latex]x[/latex]-coordinate of point [latex]P[/latex].
    • [latex]\sin(\theta)[/latex] is the [latex]y[/latex]-coordinate of point [latex]P[/latex]. 
  4. Remember that the signs of the sine and cosine are determined by the quadrant in which point [latex]P[/latex] lies:
      • In Quadrant I, both sine and cosine are positive.
      • In Quadrant II, sine is positive and cosine is negative.
      • In Quadrant III, both sine and cosine are negative.
      • In Quadrant IV, sine is negative and cosine is positive. 

The terminal side of an angle is the side that moves or rotates from the initial side to form the angle. The position of the terminal side after this rotation determines the magnitude of the angle.

Illustration of an angle with labels for initial side, terminal side, and vertex.

Find the coordinates of the point on the unit circle at an angle of [latex]\frac{5\pi }{3}[/latex].