__Complementary angles in trigonometry :__

Two acute angles are complementary to each other if their sum is equal to 90°. In a right triangle the sum of the two acute angles is equal to 90°. So, the two acute angles of a right triangle are always complementary to each other.

Let ABC be a right triangle, right angled at B.

If <ACB = θ, then <BAC = 90° - θ and hence the angles <BAC and <ACB are complementary

For the angle θ, we have

Similarly, for the angle (90° - θ), we have

Comparing the equations in (1) and (2) we get,

*Key concept* - Trigonometric Ratios of Complementary Angles

**Problem 1 :**

Evaluate : cos 56° / sin 34°

**Solution : **

The angles 56° and 34° are complementary.

So, using trigonometric ratios of complementary angles, we have

cos 56° = cos (90° - 56°) = sin 34°

cos 56° / sin 34° = sin 34° / sin 34° = 1

**Hence the value of cos 56° / sin 34° is 1.**

**Problem 2 :**

Evaluate : tan 25° / cot 65°

**Solution : **

The angles 25° and 65° are complementary.

So, using trigonometric ratios of complementary angles, we have

tan 25° = tan (90° - 65°) = cot 65°

tan 25° / cot 65° = cot 65° / cot 65° = 1

**Hence the value of tan 25° / cot 65° is 1.**

**Problem 3 :**

Evaluate : (cos 65° sin 18° cos 58°) / (cos 72° sin 25° sin 32°)

**Solution : **

Using trigonometric ratios of complementary-angles, we have

cos 65° = cos (90° - 25°) = sin 25°

sin 18° = sin (90° - 72°) = cos 72°

cos 58° = cos (90° - 32°) = sin 32°

(cos 65° sin 18° cos 58°) / (cos 72° sin 25° sin 32°) is

= (sin 25° cos 72° sin 32°) / (cos 72° sin 25° sin 32°)

= 1

**Hence the value of the given trigonometric expression is 1.**

**Problem 4 :**

Prove : tan 35° tan 60° tan 55° tan 30° = 1

**Solution : **

**Using trigonometric ratios of complementary-angles, we have**

tan 35° = tan (90° - 55°) = cot 55° = 1/tan 55°

tan 60° = tan (90° - 30°) = cot 30° = 1/tan 30°

tan 35° tan 60° tan 55° tan 30° is

= (1/tan 55° ) x (1/tan 30°) tan 55° tan 30°

= 1

**Hence, tan 35° tan 60° tan 55° tan 30° = 1**

**Problem 5 :**

If sin A = cos 33°, find A

**Solution : **

**Using trigonometric ratios of complementary-angles, we have**

sin A = cos (90° - A)

Therefore,

sin A = cos 33° -----> cos (90° - A) = cos 33°

90° - A = 33°

90° - 33° = A

57° = A

**Hence, A is 57°.**

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