CBSE Class 10 Maths: Introduction to Trigonometry Ex 8.3

Understanding Trigonometric Ratios of Complementary Angles

In a right-angled triangle, if one acute angle is 'A', then the other acute angle must be (90° - A), because the sum of all angles in a triangle is 180°, and one angle is 90°. These two acute angles, A and (90° - A), are called complementary angles as their sum is 90°.

The beauty of trigonometry is how ratios of complementary angles are related. Let's consider a right-angled triangle ABC, right-angled at B. If ∠C = A, then ∠A = 90° - A.

Now, let's look at the ratios:

• sin A = Opposite/Hypotenuse = AB/AC
• cos (90° - A) = Adjacent/Hypotenuse = AB/AC

This clearly shows that sin A = cos (90° - A). Similarly, by observing the other sides and ratios, we can derive the following fundamental identities for complementary angles:

1. sin (90° - A) = cos A
2. cos (90° - A) = sin A
3. tan (90° - A) = cot A
4. cot (90° - A) = tan A
5. sec (90° - A) = cosec A
6. cosec (90° - A) = sec A

These identities are incredibly useful for simplifying trigonometric expressions, especially when you encounter angles that are not standard values but sum up to 90°. The trick is to identify such pairs and apply the appropriate transformation. For example, if you see sin 25° and cos 65°, you immediately know 25° + 65° = 90°, so cos 65° can be written as sin (90° - 65°) = sin 25°.

Step-by-Step Problem Solving with Complementary Angles

1. Identify Complementary Angle Pairs — Look for angles in the given expression that add up to 90°. For example, (20°, 70°), (35°, 55°), etc. This is the first and most crucial step.
2. Choose One Angle to Transform — For each complementary pair, decide which trigonometric ratio you want to convert. For instance, if you have sin 20° and cos 70°, you can either change sin 20° to cos (90°-20°) = cos 70° or change cos 70° to sin (90°-70°) = sin 20°. The goal is to make the angles and/or the trigonometric functions match for simplification.
3. Apply the Correct Identity — Use the identities: sin(90°-A)=cos A, cos(90°-A)=sin A, tan(90°-A)=cot A, cot(90°-A)=tan A, sec(90°-A)=cosec A, cosec(90°-A)=sec A. Ensure you don't confuse which ratio transforms into which.
4. Simplify the Expression — After applying the identities, simplify the expression. This might involve cancellation, addition/subtraction of like terms, or using other fundamental trigonometric identities (like tan A = sin A / cos A, sec A = 1/cos A).
5. Verify (Optional but Recommended) — If time permits, especially for proving identities, quickly re-check your steps and calculations to ensure accuracy.

Worked Examples from Exercise 8.3

• Example 1: Evaluate tan 65° / cot 25° Step 1: Notice that the angles 65° and 25° are complementary, as 65° + 25° = 90°. Step 2: We can convert either tan 65° or cot 25°. Let's convert tan 65°. Using the identity tan (90° - A) = cot A, we have tan 65° = tan (90° - 25°) = cot 25°. Step 3: Substitute this into the original expression: tan 65° / cot 25° = cot 25° / cot 25° Step 4: Simplify the expression. cot 25° / cot 25° = 1 Final Answer: 1
• Example 2: If sin 3A = cos (A - 26°), where 3A is an acute angle, find the value of A. Step 1: The given equation is sin 3A = cos (A - 26°). We know that sin θ = cos (90° - θ). Step 2: Apply the identity to the left side: sin 3A = cos (90° - 3A). Step 3: Now, equate the transformed expression with the right side of the original equation: cos (90° - 3A) = cos (A - 26°) Step 4: Since the cosine of two acute angles is equal, the angles themselves must be equal. 90° - 3A = A - 26° Step 5: Solve for A. 90° + 26° = A + 3A 116° = 4A A = 116° / 4 A = 29° Step 6: Verify if 3A is acute: 3 * 29° = 87°, which is an acute angle. Final Answer: A = 29°

Common Mistakes and Exam Tips

1. Confusing Identities: A very common mistake is to mix up which ratio converts to which. Remember: sin and cos are complementary, tan and cot are complementary, and sec and cosec are complementary. Forgetting this can lead to incorrect substitutions (e.g., sin(90-A) = sec A).
2. Incorrect Angle Subtraction: When using (90° - A), ensure you subtract the angle correctly. For expressions like (A - 26°), be careful if you are transforming cos (A - 26°) = sin (90° - (A - 26°)) = sin (90° - A + 26°) = sin (116° - A).
3. Not Identifying Complementary Pairs: Sometimes students try to convert ratios where the angles are not complementary, making the problem harder or leading to wrong answers. Always check if the angles sum to 90° first.
4. Leaving Unsimplified Expressions: After applying the identities, always simplify the expression to its lowest possible form. Don't stop halfway.
5. Practice is Key: The best way to avoid these mistakes is consistent practice. Work through all the examples and exercises from your NCERT textbook and additional problems.

Practice Questions with Solutions

• Q: Evaluate: sin 18° / cos 72° A: Step 1: Notice that 18° + 72° = 90°, so they are complementary angles. A: Step 2: Convert sin 18° using the identity sin (90° - A) = cos A. So, sin 18° = sin (90° - 72°) = cos 72°. A: Step 3: Substitute this into the expression: cos 72° / cos 72°. A: Step 4: Simplify the expression. Final answer: 1
• Q: If tan 2A = cot (A - 18°), where 2A is an acute angle, find the value of A. A: Step 1: The given equation is tan 2A = cot (A - 18°). We know that tan θ = cot (90° - θ). A: Step 2: Apply the identity to the left side: tan 2A = cot (90° - 2A). A: Step 3: Equate the transformed expression with the right side: cot (90° - 2A) = cot (A - 18°). A: Step 4: Since cotangent of two acute angles is equal, the angles must be equal: 90° - 2A = A - 18°. A: Step 5: Solve for A: 90° + 18° = A + 2A => 108° = 3A => A = 108° / 3. Final answer: A = 36°
• Q: Show that tan 48° tan 23° tan 42° tan 67° = 1. A: Step 1: Group complementary angle pairs: (48°, 42°) because 48° + 42° = 90°; and (23°, 67°) because 23° + 67° = 90°. A: Step 2: Use the identity tan (90° - A) = cot A. A: Step 3: Convert tan 48° to cot (90° - 48°) = cot 42°. A: Step 4: Convert tan 23° to cot (90° - 23°) = cot 67°. A: Step 5: Substitute these into the expression: (cot 42°) (cot 67°) (tan 42°) (tan 67°). A: Step 6: Rearrange and use cot A = 1 / tan A: (cot 42° tan 42°) (cot 67° tan 67°) = (1) (1). Final answer: 1 (Hence proved)
• Q: Express cosec 68° + cos 85° in terms of trigonometric ratios of angles between 0° and 45°. A: Step 1: We need to convert 68° and 85° into angles between 0° and 45° using complementary angle identities. A: Step 2: For cosec 68°, use cosec (90° - A) = sec A. So, cosec 68° = cosec (90° - 22°) = sec 22°. A: Step 3: For cos 85°, use cos (90° - A) = sin A. So, cos 85° = cos (90° - 5°) = sin 5°. A: Step 4: Substitute these converted terms back into the original expression. Final answer: sec 22° + sin 5°

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