Trig Ratios of Complementary Angles Trigonometric Ratios of Complementary Angles

Two angles are called complementary when their measures add up to 90 degrees. So 30 and 60 are complementary, and 25 and 65 are complementary. In any right-angled triangle the angle at the right angle is 90 degrees, which leaves exactly 90 degrees to share between the other two angles. Therefore those two acute angles are always complementary to each other: if one is A, the other must be 90 - A.

Look at right triangle ABC with the right angle at B. For angle A, side a is the opposite side and side c is the adjacent side. Now switch your attention to angle C, which equals 90 - A. From C's point of view the same side a is now the adjacent side and c is the opposite side; the roles have swapped. Because sine uses opposite over hypotenuse and cosine uses adjacent over hypotenuse, the sine of one angle equals the cosine of the other. That is the whole idea behind sin(90 - A) = cos A.

Carrying the swap through all three pairs gives six relations: sin(90 - A) = cos A, cos(90 - A) = sin A, tan(90 - A) = cot A, cot(90 - A) = tan A, sec(90 - A) = cosec A and cosec(90 - A) = sec A. The pattern is simple to remember: replace the ratio by its 'co' partner and replace the angle by its complement. These hold for every acute angle A.

These relations are a workhorse in ICSE problems. They let you rewrite an expression so terms cancel, for example sin 35 / cos 55 = sin 35 / sin 35 = 1, because cos 55 = cos(90 - 35) = sin 35. They also explain why a four-figure trig table only lists angles from 0 to 45 degrees: any angle above 45 is found by reading its complement, since cos 70 is just sin 20.