How to Find sin θ and cos θ When tan θ Is Given

Can tangent give you sine and cosine?

For an acute angle θ with tan θ = p/q and p, q positive, take the opposite and adjacent sides in the ratio p:q. The hypotenuse is proportional to √(p² + q²). Therefore sin θ = p/√(p² + q²) and cos θ = q/√(p² + q²).

Worked example: tan θ = 3/4

Draw a right triangle with the side opposite θ labelled 3 and the adjacent leg labelled 4. Pythagoras gives hypotenuse √(9 + 16) = 5. Thus sin θ = 3/5 and cos θ = 4/5.

These labels represent a ratio; the actual triangle could have sides 6, 8 and 10. Multiplying every side by the same positive number leaves all three trigonometric ratios unchanged.

Check the answer in two ways

First, sin θ/cos θ = (3/5)/(4/5) = 3/4, matching the supplied tangent. Second, sin²θ + cos²θ = 9/25 + 16/25 = 1. Both checks help detect an incorrect hypotenuse.

If tan θ = 2, treat it as 2/1. The hypotenuse is √5, so sin θ = 2/√5 and cos θ = 1/√5. Rationalised forms are 2√5/5 and √5/5.

Why the angle condition matters

The positive formulas above assume an acute angle. For general angles, tangent alone does not determine the individual signs of sine and cosine. A positive tangent can occur where both are positive or where both are negative. Use the quadrant information when the question goes beyond acute-angle right triangles.

Practice with answers

• Acute θ, tan θ = 5/12: sin θ = 5/13, cos θ = 12/13.
• Acute θ, tan θ = 1: sin θ = cos θ = 1/√2.
• Acute θ, tan θ = 8/15: sin θ = 8/17, cos θ = 15/17.

Do not confuse the adjacent leg with the hypotenuse: tangent uses the two legs, while sine and cosine use the hypotenuse.

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