Madhyamik Mathematics Suggestion – Trigonometric Ratios of Complementary Angles is designed for WBBSE Class 10 students who are revising an important part of Trigonometry. This chapter focuses on the relationships between trigonometric ratios of complementary angles and provides objective, short-answer and descriptive practice.
The questions below are arranged in a clear format so that students can revise the important identities, solve typical problems and strengthen their preparation for the Madhyamik Mathematics examination.
Trigonometric Ratios of Complementary Angles
When two angles have a sum of 90°, they are called complementary angles. The trigonometric ratios of such angles are closely related to one another.
cos(90° − θ) = sin θ
tan(90° − θ) = cot θ
cot(90° − θ) = tan θ
sec(90° − θ) = cosec θ
cosec(90° − θ) = sec θ
These identities are especially useful when simplifying expressions containing angles such as 20°, 22°, 68°, 70°, 75° and their complementary angles.
Important MCQ Questions – 1 Mark
1. If sec 40° = cosec(θ − 20°), then θ is equal to:
(a) 20° (b) 21° (c) 22° (d) 25°
2. Find the value of:
(a) 1 (b) −1 (c) 2 (d) 0
3. If sin(40° + θ) = cos(50° − θ), then the value of θ is:
(a) 2θ (b) 2θ + 2 (c) 0 (d) 1
4. If tan θ · tan 40° = 1, then θ is:
(a) 12° (b) 15° (c) 18° (d) 20°
True or False Questions
1. tan 4° · tan 43° · tan 47° · tan 86° = 1.
2. sin 18° and cos 72° have the same value.
3. In a square ABCD, tan(A/2) · tan(A/2) = 1.
4. 46° and 44° are complementary angles.
5. cos²21° + cos²69° = 2.
Fill in the Blanks – 1 Mark
1. If A + B = 90° and tan A = 3/4, then cot B = ______.
2. Find the value of sin²22° + sin²68° + cot²30°.
3. If cos 12° = x, then cosec78° = ______.
4. tan15° · tan35° · tan45° · tan55° · tan75° = ______.
5. tan43° · cos47° + cos43° · sin47° = ______.
6. tan 2θ · tan 3θ = 1. If θ lies in the appropriate acute-angle range, find θ.
Short Answer Questions – 2 Marks
Question 1
If cosec θ = cosec φ, where θ and φ are acute angles, prove that cosec(θ + φ) = 1.
Question 2
If
find the value of tan θ.
Answer: tan θ = 4/3
Important Trigonometric Identities for Revision
| Expression | Equivalent Form |
|---|---|
| sin(90° − θ) | cos θ |
| cos(90° − θ) | sin θ |
| tan(90° − θ) | cot θ |
| cot(90° − θ) | tan θ |
| sec(90° − θ) | cosec θ |
| cosec(90° − θ) | sec θ |
More Short Problems with Solutions
Problem 1: Simplify tan15° + tan75°
Therefore, the required expression can be simplified using the complementary-angle identity.
Problem 2: Simplify cosec²22° · cot²68°
Answer: sec²22°
Long Answer Questions – 5 Marks
Question 1
Prove the required trigonometric relation using the complementary-angle identities involving 15° and 75°.
The key step is to recognise that the two angles are complementary.
Question 2
Prove the identity involving sin²22°, sin²68° and cot²30°.
Hence proved.
Question 3
If tan α = cot β, where α and β are acute angles, prove that cos(α + β) = 0.
Hence proved.
Key Points to Remember
- Two angles whose sum is 90° are complementary angles.
- sin of an angle is equal to cos of its complementary angle.
- tan of an angle is equal to cot of its complementary angle.
- sec of an angle is equal to cosec of its complementary angle.
- Whenever angles such as 22° and 68° or 15° and 75° appear together, check whether they are complementary.
- For simplification, use standard identities instead of calculating decimal values.
- In long-answer questions, write each transformation step clearly to make the solution easy to follow.
Quick Revision Table
| Pair of Angles | Important Relation |
|---|---|
| 20° and 70° | 20° + 70° = 90° |
| 22° and 68° | 22° + 68° = 90° |
| 30° and 60° | 30° + 60° = 90° |
| 15° and 75° | 15° + 75° = 90° |
| 40° and 50° | 40° + 50° = 90° |
| 43° and 47° | 43° + 47° = 90° |
How to Prepare This Chapter for Madhyamik Mathematics
Start by learning the six complementary-angle relationships. Once these identities become familiar, practise objective questions involving pairs such as 22°–68°, 15°–75° and 43°–47°.
For short and long questions, avoid jumping directly to the final answer. Write the identity first, substitute the complementary angle and simplify the expression step by step. This approach makes your calculation easier to check and improves answer presentation.
Conclusion
Trigonometric Ratios of Complementary Angles is an important practice area in Madhyamik Mathematics. The main focus should be on recognising complementary angles and applying the correct trigonometric relationship without unnecessary calculation.
Revise the identities, practise the MCQs and fill-in-the-blanks, and then move to short and long-answer problems. Regular written practice will help improve both accuracy and speed when solving trigonometric questions.