Complementary Angle Trigonometric Ratios

Quick answer: For complementary angles, sin(90° − θ) = cos θ and cos(90° − θ) = sin θ. Also tan(90° − θ) = cot θ where both are defined.

Worked example

Simplify sin 35°/cos 55° and sin²20° + sin²70°.

1. Since 55° = 90° − 35°, cos 55° = sin 35°; the first expression equals 1.
2. Since 70° = 90° − 20°, sin 70° = cos 20°.
3. The second becomes sin²20° + cos²20° = 1.

Check the result

Both transformations replace a ratio by its cofunction at the complementary angle.

Common mistake to avoid

The identity uses 90° in degrees, or π/2 in radians. Do not replace sin(90° − θ) by sin θ.

Try it yourself

1. Simplify cos 25° − sin 65°. 2. Evaluate tan 30° × tan 60°.

Answers and reasoning

1. Zero, because the two ratios are equal. 2. (1/√3)(√3) = 1.

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