Using the Unit Circle

To use the Unit Circle to evaluate cosine, sine, or tangent we use the coordinates of the point of intersection between the terminal side of the angle and the Unit Circle.

The x-coordinate of the point is equal to the cosine of the angle.

The y-coordinate of the point is equal to the sine of the angle.

To find the tangent of an angle you take the y-coordinate/x-coordinate.

Cos(angle) = x-coordinate of point

Cosine is positive in the FIRST and FOURTH quadrants.

Cosine is negative in the SECOND and THIRD quadrants.

cos(2π/3) = -1/2

cos(270°) = 0

cos(π/6) = √3/2

Sin(angle) = y-coordinate of point

Sine is positive in the FIRST and SECOND quadrants.

Sine is negative in the THIRD and FOURTH quadrants.

sin(135°) = √2/2

sin(4π/3) = -√3/2

sin(0) = 0

Tan(angle) = y-coordinate/x-coordinate

Tangent is positive in the FIRST and THIRD quadrants.

Tangent is negative in the SECOND and FOURTH quadrants.

tan(90°) = 1/0 = UNDEFINED

tan(π) = 0/-1 = 0

tan(240°) = (-√3/2)/(-1/2)

=(-√3/2) *(-2/1) = √3/1 = √3