The key idea with triangles who's hypotenuse lie on the unit circle (that form an angle of θ with the x-axis) is that cosθ represents length of the base, and sinθ represents the height.

Take a look at the graph below and notice the following relationships always hold:

The following table shows the values of sinθ and cosθ for the so called critical angles θ. These are angles that give "nice" values for sin and cos:

Once the values of sin and cos are memorized for critical angles in the first quadrant, you can use the rules of symmetry and periodicity to determine the values of critical angles in arbitrary quadrants.

The unit circle can be divided into quadrants based on the sign of cosθ and sinθ. These correspond to the 4 quadrants produced by the intersection of the x and y axes. The quadrants are denoted Q1, Q2, Q3 and Q4.

Since a full circle is 2π radians, adding 2π to any angle θ gives the same point on the unit circle. In fact, adding any integer multiple of 2π gives the same point:

A third trigonometric function, the tangent, is defined by

Since sin and cos are respectively the y and x coordinates of a point on the unit circle, tan is the ratio of these - aka the slope of the line from the origin to (cosθ,sinθ).

Example

Find the exact value of tan(617π​).

Remember that cosθ, sinθ are the base and height of a right angled triangle whose hypotenuse is 1, we can apply Pythagoras's Theorem and conclude:

Example

Given that cosθ=−53​ and π≤θ≤2π, find sinθ.

And since π≤θ≤2π, sinθ<0 so sinθ=−54​.