Precalculus - The Unit Circles


  • A Unit Circle is a type of circle whose center is at 0, 0 and whose radius length is equal to 1 unit. 
  • This circle is modeled by the equation x2+y2=1. This equation is derived using the distance formula.

  • The circumference of the unit circle is equal to 2π units or approximately equal to 6.28 units.
  • To inspect whether a point x, y is on the unit circle, we check the coordinates if they satisfy the equation x2+y2=1.
  • The standard arc on the circle starts at the point 1, 0 and travels counterclockwise if it is positive and clockwise if it is negative.
  • Since the circumference of the unit circle is 2π, then the arc lengths are rational multiples of 2π
  • Consider the unit circle being divided by the coordinate axes into four congruent arcs.

  • As point X moves from point A counterclockwise and terminates at point B, the length of arc AB is 14 of 2π or π2.
  • The length of arc ABC is one-half of the circumference, which is π.
  • The length of arc ACD is 34 of the circumference, which is 3π2.
  • If we consider θ as the length of the arc, then we have the following:
    •  Xθ=Xπ2=0, 1
    • Xθ=Xπ=-1,0
    • X(θ)=X(3π2)=(0,-1) 
    •  Xθ=X2π=1, 0

Properties of the Unit Circle and the Special Angles

  • As shown in the figure, each point on the unit circle satisfies the equation x2+y2=1.
  • Each point on the circle has coordinates corresponding to a special angle or multiple of special angles, such as 30°, 60°, and 90°.
  • Study the tables below.

  • The domain of the unit circle x2+y2=1 is x|x,-1x1. The range is y|y,-1y1.
  • Every real number (arc length) is associated uniquely with the central angle θ that subtends the standard arc on the unit circle.
  • The relationship Pθ+2kπ=Pθ, for any integer k is called the Periodic Theorem. Typical points on the unit circle range from 0° to 360°. Once the angle of the circular function exceeds 360°, then this multiple of 360° has to be subtracted from the given angle. The difference is then equivalent to one common value between 0° to 360°.

Trigonometric Functions and the Unit Circle

  • Let us consider a real number θ and let Px, y be a point on the unit circle corresponding to the angle θ.
  • For each arc length on the unit circle, with the starting point 1, 0 and terminal point x, y, the values of x and y are real numbers, then we have cosθ=x and sinθ=y.
  • Study the figure below.

  • Point Px,y is equal to Pcosθ,sinθ. This indicates that sinθ=y and cosθ=x. This is based on sinθ=yr=y1=y and cosθ=xr=x1=x
  • Using the y and x coordinates, we can define the other trigonometric functions of θ.
  • tanθ=yx, x0cotθ=xy, y0, secθ=1x, x0, and cscθ=1y, y0.
  • Trigonometric Functions are also called Circular Functions or Periodic Functions because they behave in a cyclic or repetitive manner.
  • To illustrate, we have sinθ+2π=sinθ and cosθ+2π=cosθ as these functions have a period of 2π.


Trigonometric Functions of Any Angle

  • If we let θ be any angle in standard position, it indicates that its vertex is at the origin and the initial side is the positive x-axis.
  • The six trigonometric functions are described in the tables below, and their respective signs in the four quadrants.

Reference Angle

  • For any angle θ, its reference angle is the acute angle that the terminal side of θ makes with the positive x-axis or negative x-axis.
  • If 0°<θ<90°, then the reference angle β is equal to θ.
  • In Quadrant II, the reference angle is β=180°-θ.
  • In Quadrant III, the reference angle is β=θ-180°.
  • In Quadrant IV, the reference angle is β=360°-θ.

In general, the angle θ and θ+360°n where n have the same terminal side. These angles are called co-terminal angles.

Solved Examples

Example 1. Find the exact value of cos13π6 using the standard unit circle.



13π6 terminates in Quadrant I, which forms 30° in the positive x-axis.

Consider the angle 30° and choose the point 32,12, on the unit circle.

This shows that cos13π6=cosπ6=32

Note that the sign is positive since cosθ>0 in Quadrant I.


Example 2. Given that tanθ=512 and sinθ<0, find the other trigonometric functions of θ.


If the tangent function is positive and sine function is negative, then θ is Quadrant III.


Solve for the radius:



sinθ=-513, cosθ=-1213, cotθ=125, secθ=-1312, cscθ=-135

Cheat Sheet

  • Trigonometric functions of different special angles are summarized in the tables below.

  • The cosine function and secant functions are even. To illustrate, we have cos-θ=cosθ and sec-θ=secθ.
  • The sine, cosecant, tangent, and cotangent functions are odd.

Blunder Areas

  • Always be mindful of the sign of each trigonometric function in the four quadrants.
  • Reference angles can be used to evaluate trigonometric functions.
  • Suppose that tanθ=-13 and we want to find the other trigonometric values of θ, we solve for the radius. The radius is always positive.
  • The angles associated with trigonometry (in the unit circle approach) are directed angles. Angles obtained from the counterclockwise direction are positive while angles obtained from the clockwise direction are negative.