Trigonometric ratios in each quadrant

At angle θ, measured anticlockwise from the positive horizontal axis, the unit-circle point has coordinates (cos θ, sin θ). Cosine is positive on the right half of the circle and sine is positive on the upper half. Tangent is sin θ/cos θ where cosine is nonzero, so its sign follows both coordinate signs.

A reference angle gives the positive magnitudes from a first-quadrant triangle. Attach the signs from the actual quadrant afterwards. Sine is positive in quadrants one and two; cosine in one and four; tangent in one and three. Values at an axis need separate attention: tangent is undefined at 90° and 270°.

Worked example

Find sin 150°, cos 150° and tan 150° exactly.

1. 150° is in quadrant two with reference angle 30°.
2. Sine is positive and cosine is negative there.
3. Use tan θ = sin θ/cos θ and rationalise the denominator.

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

Revise first: Exact trigonometric values.