Trigonometric Ratios of Complementary Angles

Complementary angles are two angles that add up to 90°. For example, 30° and 60° are complementary angles. In this article, we will learn what complementary angles are and how they are used in trigonometry.

Table of Contents

• Trigonometric Ratio of Complementary Angles
• Finding Trigonometric Ratios of Complementary Angles
• Solved Examples on Trigonometric Ratios of Complementary Angles
• Practice Questions on Trigonometric Ratios of Complementary Angles
• Conclusion

Trigonometric Ratio of Complementary Angles

In a right triangle, two angles are always complementary to each other, meaning they add up to 90°. When we look at the trigonometric ratios of these angles, we can see a clear and simple pattern between them.

If A and B are two complementary angles, then: A + B = 90°, which means B = 90° − A.

The six trigonometric ratios of complementary angles are:

Trigonometric Ratio	Complementary Angle Relation
sin A	= cos (90° − A)
cos A	= sin (90° − A)
tan A	= cot (90° − A)
cot A	= tan (90° − A)
sec A	= cosec (90° − A)
cosec A	= sec (90° − A)

These relations show that sine and cosine, tangent and cotangent, and secant and cosecant are pairs. Each one is the complementary ratio of the other.

Finding Trigonometric Ratios of Complementary Angles

To find the trigonometric ratio of a complementary angle, follow these simple steps:

Step 1: Check the given angle and the trigonometric function.

Step 2: Use the identity to convert it. For example, sin(60°) = cos(90° − 60°) = cos(30°).

Step 3: Write the final value using a standard table or known values.

Here are a few quick conversions to remember:

• sin(30°) = cos(60°) = 1/2
• sin(45°) = cos(45°) = 1/√2
• sin(60°) = cos(30°) = √3/2
• tan(30°) = cot(60°) = 1/√3
• tan(45°) = cot(45°) = 1
• tan(60°) = cot(30°) = √3

These values come directly from the complementary angle relationship and are very useful in solving problems quickly.

Solved Examples

Example 1: Find the value of sin(70°) using complementary angles.

Solution:

We know that sin A = cos(90° − A)

So, sin(70°) = cos(90° − 70°) = cos(20°)

Example 2: Simplify: tan(35°) / cot(55°)

Solution:

We know that cot(55°) = cot(90° − 35°) = tan(35°)

So, tan(35°) / cot(55°) = tan(35°) / tan(35°) = 1

Example 3: If sin(3A) = cos(A − 10°), find the value of A.

Solution: Since sin(3A) = cos(90° − 3A), we can write:

cos(90° − 3A) = cos(A − 10°)

So, 90° − 3A = A − 10°

90° + 10° = A + 3A

100° = 4A

A = 25°

Example 4: Find the value of: sin(65°) / cos(25°)

Solution:

We know that cos(25°) = cos(90° − 65°) = sin(65°)

So, sin(65°) / cos(25°) = sin(65°) / sin(65°) = 1

Example 5: Show that tan(48°) × tan(42°) = 1

Solution:

We know that tan(42°) = tan(90° − 48°) = cot(48°)

So, tan(48°) × tan(42°) = tan(48°) × cot(48°)

Since tan × cot = 1,

tan(48°) × tan(42°) = 1

Practice Questions

Try solving these questions on your own to test your understanding:

1. Find the value of cos(80°) using complementary angle identity.
2. Simplify: sin(50°) / cos(40°)
3. If cos(2A) = sin(A + 15°), find the value of A.
4. Prove that: sin(30°) × sec(60°) = 1
5. Find the value of: tan(27°) × tan(63°)
6. Simplify: (sin 35° × cos 55°) + (cos 35° × sin 55°)
7. If tan(4A) = cot(A − 5°), find the value of A.
8. Show that: cosec(72°) − sec(18°) = 0

Conclusion

Complementary angles and their trigonometric ratios are a simple yet powerful concept in mathematics. Once you understand the basic pairs  sin/cos,  tan/cot, and  sec/cosecsolving problems becomes much faster and easier.