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 rations of compound angles.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

ALGEBRA

Variables and constants

Writing and evaluating expressions

Solving linear equations using elimination method

Solving linear equations using substitution method

Solving linear equations using cross multiplication method

Solving one step equations

Solving quadratic equations by factoring

Solving quadratic equations by quadratic formula