# TRIGONOMETRIC RATIOS FOR COMPLEMENTARY ANGLES

## About "Trigonometric ratios for complementary angles"

Trigonometric ratios for 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

 sin θ =  AB/ACcos θ  =  BC/ACtan θ  =  AB/BC cosec θ =  AC/ABsec θ  =  AC/BCcot θ  =  BC/AB

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

 cos (90 - θ) =  AB/ACsin (90 - θ)  =  BC/ACtan (90 - θ)  =  BC/AB cosec (90 - θ)  =  AC/ABsec (90 - θ)  =  AC/BCcot (90 - θ)  =  AB/BC

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

sin θ  =  AB/AC  =  cos (90-θ)

cos θ  =  BC/AC  =  sin (90-θ)

tan θ  =  AB/BC  =  cot (90-θ)

cosec θ  =  AC/AB  =  sec (90-θ)

sec θ  =  AC/BC  =  cosec (90-θ)

cot θ  =  BC/AB  =  tan (90-θ)

## Trigonometric ratios of complementary angles

 sin θ  =  cos (90 - θ)cos θ  =  sin (90 - θ)tan θ  =  cot (90 - θ) cosec θ  =  sec (90 - θ)sec θ  =  cosec (90 - θ)cot θ  =  tan (90 - θ)

## Trigonometric ratios of complementary angles - Practice problems

Example 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.

Example 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.

Example 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 :

Example  tan 35° tan 60° tan 55° tan 30°   =  1

Solution :

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

Example 5 :

If sin A  =  cos 33°, find A

Solution :

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°.

Let us look into the next example on "Trigonometric ratios for complementary angles"

Example 6 :

If tan A tan 35°  =  1, find A

Solution :

tan A tan 35°  =  1

tan A  =  1/tan 35°

tan A  =  cot 35°

cot (90 - A)  =  cot 35°

90 - A  =  35

A  =  90 - 35  =  55

Hence the value of A is 55°

After having gone through the stuff given above, we hope that the students would have understood "Trigonometric ratios for complementary angles"

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