Trigonometric Ratios of Complementary Angles: Formula, Proof, Examples and Questions

Trigonometric ratios of complementary angles describe the relationship between the trigonometric functions of two angles whose sum is 90°. These identities make it easier to simplify expressions, solve trigonometric equations and find unknown angles without doing long calculations. In this article, you’ll explore the six complementary angle identities, understand their proofs and work through solved examples and practice questions designed for CBSE Class 10 students.

Table of Contents

• What are Trigonometric Ratio of Complementary Angles?
• Formula for Trigonometric Ratios of Complementary Angles
• Finding Trigonometric Ratios of Complementary Angles
• Proof of Trigonometric Ratios of Complementary Angles
• Solved Examples on Trigonometric Ratios of Complementary Angles
• Practice Questions on Trigonometric Ratios of Complementary Angles
• Conclusion
• Frequently Asked Questions on Trigonometric Ratios of Complementary Angles

What are 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.

Formula for Trigonometric Ratios of Complementary Angles

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.

Proof of Trigonometric Ratios of Complementary Angles

Consider the right-angled triangle ABC given above, where:

• ∠B = 90°
• ∠A = θ
• ∠C = (90° − θ)

Since the sum of the angles in a triangle is 180°,

∠A + ∠B + ∠C = 180°

Substituting ∠B = 90°,

∠A + ∠C = 90°

Hence, ∠A and ∠C are complementary angles.

Proof of sin⁡A=cos⁡(90° − A)

From the definition of trigonometric ratios,

For angle A,

sin⁡A = $\frac{\text{Opposite}}{\text{Hypotenuse}} =\frac{BC}{AC}$​

For angle C,

cos⁡C =  $\frac{\text{Adjacent}}{\text{Hypotenuse}} =\frac{BC}{AC}$​

Since C = 90° − A

we get sin⁡A=cos⁡(90°−A)

Similarly,

For angle A,

$cos⁡A = \frac{AB}{AC}​$

For angle C,

 $sin⁡C = \frac{AB}{AC}$​

Therefore, cos⁡A = sin⁡(90°−A)

Using the same right triangle, we can similarly prove the remaining complementary angle identities

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

Trigonometric ratios of complementary angles help simplify calculations by converting one trigonometric function into its complementary function. Remember that sine pairs with cosine, tangent pairs with cotangent and secant pairs with cosecant. Once you master these identities, solving Class 10 trigonometry questions becomes much quicker and easier.

Know more about related topics:

• Sin Cos Tan Values
• Trigonometry Table