Unit Circle

Formulas

• Point on the circle (cos θ, sin θ)
• Tangent tan θ = sin θ / cos θ
• Identity sin²θ + cos²θ = 1
• Degrees to radians multiply by π / 180

Worked example

Find the exact values of sin, cos and tan at 150°.

1. 150° is in the second quadrant, where sin is positive and cos is negative.
2. The reference angle is 180° − 150° = 30°.
3. sin 30° = 1/2 and cos 30° = √3/2.
4. So sin 150° = 1/2, cos 150° = −√3/2 and tan 150° = (1/2) / (−√3/2) = −1/√3 = −√3/3.

Answer: sin 150° = 1/2, cos 150° = −√3/2, tan 150° = −√3/3.

Common mistakes

• Giving cos a positive sign in the second quadrant: only sin is positive there.
• Swapping sin and cos: x is cos θ and y is sin θ.
• Writing a value for tan 90°: it is undefined, because cos 90° = 0.
• Forgetting which unit the question uses: 150° is 5π/6 radians.

About this tool

The unit circle is a circle of radius 1 centred on the origin. For any angle θ measured anticlockwise from the positive x-axis, the point on the circle is (cos θ, sin θ), and tan θ = sin θ / cos θ. Drag the point and this unit circle shows the angle in degrees and radians, the right-angled triangle under it, the exact values of sin, cos and tan at the special angles (such as √3/2 at 30° and 60°) and the sine and cosine graphs tracing out as you go.

Tips

• CAST tells you which ratio is positive in each quadrant: all in the first, sin in the second, tan in the third, cos in the fourth.
• To turn degrees into radians, multiply by π/180. So 30° = π/6 and 135° = 3π/4.
• tan θ is undefined at 90° and 270° because cos θ is 0 there, and you cannot divide by 0.

FAQ

Why is the point at (cos θ, sin θ)?

The radius is the hypotenuse of the right-angled triangle and it has length 1. The horizontal side is 1 × cos θ and the vertical side is 1 × sin θ, so those are the point's coordinates.

Where do exact values like √3/2 come from?

From two special triangles. Half of an equilateral triangle with sides 2 gives sides 1, √3 and 2, so sin 60° = √3/2 and cos 60° = 1/2. A right-angled isosceles triangle with legs 1 has hypotenuse √2, so sin 45° = cos 45° = 1/√2 = √2/2.

What is a reference angle?

It is the acute angle between the radius and the x-axis. An angle and its reference angle have the same sin, cos and tan apart from the sign, which depends on the quadrant.