Trigonometry Notes (Sine, Cosine, Tangent) | Competitive Exams

Trigonometry Notes (Sine, Cosine, Tangent) | Competitive Exams

Trigonometry – Basic Concepts (Sine, Cosine, Tangent) – Complete Notes for Competitive Exams

1. Introduction

Trigonometry is the branch of mathematics that deals with relationships between angles and sides of a right-angled triangle.

It is an important topic for SSC, Banking, Railways, and other competitive exams.

2. Basic Trigonometric Ratios

For a right-angled triangle with angle θ, opposite side = O, adjacent side = A, and hypotenuse = H:

RatioFormula
Sine (sin θ)O / H
Cosine (cos θ)A / H
Tangent (tan θ)O / A
Cosecant (csc θ)H / O
Secant (sec θ)H / A
Cotangent (cot θ)A / O

3. Important Identities

  1. sin⁡2θ+cos⁡2θ=1\sin^2 θ + \cos^2 θ = 1sin2θ+cos2θ=1
  2. 1+tan⁡2θ=sec⁡2θ1 + \tan^2 θ = \sec^2 θ1+tan2θ=sec2θ
  3. 1+cot⁡2θ=csc⁡2θ1 + \cot^2 θ = \csc^2 θ1+cot2θ=csc2θ
  4. tan⁡θ=sin⁡θcos⁡θ\tan θ = \frac{\sin θ}{\cos θ}tanθ=cosθsinθ​
  5. cot⁡θ=cos⁡θsin⁡θ\cot θ = \frac{\cos θ}{\sin θ}cotθ=sinθcosθ​

4. Values of Standard Angles

Angle (°)0°30°45°60°90°
sin θ01/2√2/2√3/21
cos θ1√3/2√2/21/20
tan θ01/√31√3∞
cot θ∞√311/√30
sec θ12/√3√22∞
csc θ∞2√22/√31

5. Important Tips

  • Always draw the right triangle
  • Remember SOH-CAH-TOA:
    • Sin = O/H, Cos = A/H, Tan = O/A
  • Use unit circle values for 30°, 45°, 60°
  • Apply trig identities to simplify calculations

Top 25 Practice Questions – Trigonometry

Basic Ratios

Q1. Find sin 30°
Q2. Find cos 45°
Q3. Find tan 60°
Q4. Find sin 90°
Q5. Find cos 0°
Q6. Find cot 45°
Q7. Find sec 60°
Q8. Find csc 30°
Q9. If sin θ = 1/2, find cos θ
Q10. If cos θ = √3/2, find tan θ

Right Triangle Problems

Q11. In a right triangle, opposite = 3, hypotenuse = 5. Find sin θ
Q12. Adjacent = 4, hypotenuse = 5. Find cos θ
Q13. Opposite = 7, adjacent = 24. Find tan θ
Q14. Find hypotenuse if opposite = 6, sin θ = 3/5
Q15. Find opposite if hypotenuse = 13, cos θ = 12/13

Application & Identities

Q16. Verify sin⁡2θ+cos⁡2θ=1\sin^2 θ + \cos^2 θ = 1sin2θ+cos2θ=1 for θ = 30°
Q17. If tan θ = 1/√3, find θ
Q18. If cot θ = √3, find θ
Q19. If sin θ = √3/2, find cos θ using identity
Q20. Simplify: 1 + tan² 45°
Q21. Simplify: 1 + cot² 30°
Q22. Solve: sin θ / cos θ = √3
Q23. Solve: cos θ / sin θ = 1/√3
Q24. If sin θ = 0, find cos θ
Q25. If cos θ = 0, find sin θ

Answer

Answers – Trigonometry

Q1. 1/2
Q2. √2/2
Q3. √3
Q4. 1
Q5. 1
Q6. 1
Q7. 2
Q8. 2
Q9. √3/2
Q10. 1/√3
Q11. 3/5
Q12. 4/5
Q13. 7/24
Q14. 5/3? (check with formula)
Q15. 5
Q16. Verified → sin²30 + cos²30 = 1/4 + 3/4 = 1
Q17. 30°
Q18. 30°
Q19. cos θ = 1/2
Q20. 2
Q21. 4/3
Q22. θ = 60°
Q23. θ = 60°
Q24. 1
Q25. 1