Trigonometry Formulas
⚡ Trigonometry Quick-Sheet for Railway Exams
🔑 Must-Know Ratios (SOH-CAH-TOA Hindi)
| Ratio | Formula | Memory Code |
|---|---|---|
| sin θ | P/H | “Pandit” |
| cos θ | B/H | “Badri” |
| tan θ | P/B | “Prasad” |
| csc θ | H/P | “Har” |
| sec θ | H/B | “Bhole” |
| cot θ | B/P | “Pandit” |
🔁 Reciprocal Identities
- csc θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ
🔄 Sign Rule (ASTC) – “All School Teachers Care”
| Quad | Positive Ratios |
|---|---|
| I | All |
| II | sin & csc |
| III | tan & cot |
| IV | cos & sec |
📐 0°, 30°, 45°, 60°, 90° Table (Fingers Trick)
Hold your right hand: thumb = 0°, little finger = 90°; angle finger → √finger/2
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | ∞ |
🧮 Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
🔄 Complementary Angles (90°–θ)
- sin(90–θ) = cos θ
- cos(90–θ) = sin θ
- tan(90–θ) = cot θ
📏 Degree–Radian Swap
- π rad = 180°
- 1° = π/180 rad
- 1 rad = 180°/π ≈ 57.3°
🔺 Small-Angle Approx (θ in rad)
- sin θ ≈ θ
- tan θ ≈ θ
- cos θ ≈ 1 – θ²/2
🧠 Mnemonics
- SOH-CAH-TOA → already Hindi-coded above
- “Pandit Badri Prasad Har Bhole Pandit” fills ratios order
- ASTC → “All School Teachers Care” for sign quadrants
🚀 Rapid-Fire MCQs – Click to Expand
-
sin 30° = ?
A) 0 B) 1 C) ½ D) √3/2
Ans: C -
If tan θ = 3/4, then sin θ is
A) 4/5 B) 3/5 C) 5/3 D) 5/4
Ans: B -
cos(90° – θ) =
A) sin θ B) –sin θ C) cos θ D) –cos θ
Ans: A -
sec²θ – tan²θ equals
A) 0 B) 1 C) –1 D) 2
Ans: B -
Which ratio is positive in III quadrant?
A) sin B) cos C) tan D) csc
Ans: C -
csc θ equals
A) 1/sin θ B) 1/cos θ C) sin θ D) cos θ
Ans: A -
rad 90° is
A) π/2 B) π C) 2π D) π/4
Ans: A -
If sin θ = 1/2, θ in I quad is
A) 30° B) 45° C) 60° D) 90°
Ans: A -
tan 45° + cot 45° =
A) 0 B) 1 C) 2 D) √2
Ans: C -
cos 0° =
A) 0 B) 1 C) ½ D) –1
Ans: B
BETA
