Class 10 Maths Introduction to Trigonometry – Formulas & NCERT Solutions
Class 10 Maths Chapter 8: Introduction to Trigonometry
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Introduction
Trigonometry studies the relationship between sides and angles of a right triangle. It is essential for higher mathematics, physics, and engineering applications.
Basic Rules and Formulas
- sin θ = Perpendicular / Hypotenuse
- cos θ = Base / Hypotenuse
- tan θ = Perpendicular / Base
- cosec θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
Examples
Example 1: If sin A = 3/5, find cos A.
sin²A + cos²A = 1 → (3/5)² + cos²A = 1 → cos²A = 16/25 → cos A = 4/5
25 Questions with Solutions
Q1. Find sin 30° and cos 60°.
sin 30° = 1/2, cos 60° = 1/2
Q2. Find tan 45°.
tan 45° = 1
Q3. If cos θ = 4/5, find sin θ.
sin²θ = 1 - cos²θ = 1 - 16/25 = 9/25 → sin θ = 3/5
Q4. Find sin 0° and cos 90°.
sin 0° = 0, cos 90° = 0
Q5. Prove sin²A + cos²A = 1.
Using Pythagoras theorem in right triangle, sin²A + cos²A = 1
Q6. Find sec 60°.
sec θ = 1/cos θ → sec 60° = 1 / (1/2) = 2
Q7. Find cosec 30°.
cosec θ = 1/sin θ → cosec 30° = 1 / (1/2) = 2
Q8. Find cot 45°.
cot θ = 1 / tan θ → cot 45° = 1/1 = 1
Q9. Prove 1 + tan²A = sec²A.
Divide sin²A + cos²A = 1 by cos²A → 1 + tan²A = sec²A
Q10. Prove 1 + cot²A = cosec²A.
Divide sin²A + cos²A = 1 by sin²A → 1 + cot²A = cosec²A
Q11. If sin A = 4/5, find cos A and tan A.
cos²A = 1 - sin²A = 1 - 16/25 = 9/25 → cos A = 3/5 → tan A = 4/3
Q12. Find sin 60° and cos 30°.
sin 60° = √3/2, cos 30° = √3/2
Q13. Find tan 30°.
tan 30° = 1 / √3
Q14. Prove sin(A + B) formula for angles A = 30°, B = 45°.
sin(A + B) = sin A cos B + cos A sin B → sin 75° = sin30°cos45° + cos30°sin45° = (1/2)(√2/2) + (√3/2)(√2/2) = (√2 + √6)/4
Q15. Height of a pole problem using tan 45°.
If tan θ = height/distance, tan 45° = 1 → height = distance
Q16. If tan A = 3/4, find sin A and cos A.
tan A = 3/4 → opposite = 3k, adjacent = 4k, hypotenuse = 5k → sin A = 3/5, cos A = 4/5
Q17. Find sin 90° - θ.
sin(90° - θ) = cos θ
Q18. Find cos 90° - θ.
cos(90° - θ) = sin θ
Q19. Evaluate sin²30° + cos²60°.
sin²30° + cos²60° = (1/2)² + (1/2)² = 1/4 + 1/4 = 1/2
Q20. Solve tan²45° + 1.
tan²45° + 1 = 1² + 1 = 2
Q21. Prove sin²A/(1 - cos A) = 1 + cos A.
sin²A/(1 - cos A) = (1 - cos²A)/(1 - cos A) = (1 - cos A)(1 + cos A)/(1 - cos A) = 1 + cos A
Q22. Find cot²θ if tan²θ = 3.
cot²θ = 1/tan²θ = 1/3
Q23. Solve sin²A + sin²B = 1, A = 30°.
sin²30° + sin²B = 1 → (1/2)² + sin²B = 1 → 1/4 + sin²B = 1 → sin²B = 3/4 → B = 60°
Q24. Find sin 45° × cos 45°.
sin 45° × cos 45° = (√2/2) × (√2/2) = 1/2
Q25. Solve a height problem using tan θ = h/d.
If tan 60° = h/√3 → h = √3 × √3 = 3 units
15 FAQs on Introduction to Trigonometry
Q1. What is Trigonometry?
Study of sides and angles of triangles.
Q2. Who is the father of Trigonometry?
Hipparchus.
Q3. Name the six trigonometric ratios.
sin, cos, tan, cosec, sec, cot.
Q4. What is sin 90°?
1
Q5. What is cos 0°?
1
Q6. Reciprocal of sin?
cosec
Q7. Write Pythagoras identity.
sin²θ + cos²θ = 1
Q8. Relation between tan and cot?
tan θ × cot θ = 1
Q9. Complementary angles?
Two angles sum to 90°
Q10. Value of tan 60°?
√3
Q11. Can trig ratios be negative?
Yes, depending on the quadrant.
Q12. Value of sec 45°?
√2
Q13. How is trigonometry used daily?
Measuring heights, distances, navigation, architecture.
Q14. Angle of elevation?
Angle above horizontal.
Q15. Angle of depression?
Angle below horizontal.
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