medium difficulty5 min

Trig ratios for complementary angles

Trigonometric ratios of complementary angles are related by swapping sine with cosine, tangent with cotangent, and secant with cosecant.

In one line: Complementary angles swap the trig pair like a mirror.

Practice This Concept

Understand it

Complementary angles add up to 90 degrees. In a right triangle, if one acute angle is A, the other is 90 degrees minus A. Because the two acute angles are complementary, the ratios linked to one angle become the complementary ratios of the other angle.

Example

sin(90 degrees - A) = cos A and tan(90 degrees - A) = cot A.

Formula and rule

sin(90 degrees - A) = cos A, cos(90 degrees - A) = sin A, tan(90 degrees - A) = cot A, cot(90 degrees - A) = tan A, sec(90 degrees - A) = cosec A, cosec(90 degrees - A) = sec A.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1If A = 30 degrees, then cos 60 degrees = sin 30 degrees = 1/2.
2This follows directly from the complementary-angle rule.

Method you can reuse

1Check whether the angle is written as 90 degrees minus something.
2Swap sine with cosine or tangent with cotangent as needed.
3Substitute the known value.
4Simplify the answer.

Sketch it in your answer

Use a right triangle and label the two acute angles A and 90 degrees minus A. The sides opposite one angle become adjacent to the other angle.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the value of cos(90 degrees - A) in terms of angle A?

Think first, then show answer

cos(90 degrees - A) equals sin A. Complementary angles swap sine and cosine.

Common wrong answer

A common wrong answer is to keep the same ratio name and only change the angle. That breaks the complementary-angle relationship.

Don't mix this up

Students often use the same ratio on both sides of the identity instead of switching to the complementary pair.

Exam tip

Whenever you see 90 degrees minus an angle, look for the complementary ratio, not the same one.