TRIGONOMETRIC RATIOS OF COMPLEMENTARY ANGLES

Trigonometric Ratios of Complementary Angles :

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 (1) and 2),

Key Concept - Trigonometric Ratios of Complementary Angles

Trigonometric Ratios of Complementary Angles - Practice Problems

Problem 1 :

Evaluate :

cos 56° / sin 34°

Solution :

The angles 56° and 34° are complementary.

Using trigonometric ratios of complementary angles,

cos 56° = cos (90° - 56°) = sin 34°

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

So, the value of cos 56° / sin 34° is 1.

Problem 2 :

Evaluate :

tan 25° / cot 65°

Solution :

The angles 25° and 65° are complementary.

Using trigonometric ratios of complementary angles,

tan 25° = tan (90° - 65°) = cot 65°

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

So, 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,

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

So, 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,

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

So,

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,

sin A = cos (90° - A)

Then,

sin A = cos 33°

cos (90° - A) = cos 33°

90° - A = 33°

90° - 33° = A

57° = A

So, A is 57°.

After having gone through the stuff given above, we hope that the students would have understood, how to solve problems using trigonometric ratios of compound angles.

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