Trigonometric Ratios & the Standard Table

🎯 Learning Objective

Define sin, cos, tan via SOH-CAH-TOA and recall their values at 0°, 30°, 45°, 60°, 90°.

💡 Concept

• In a right triangle: sin θ = Perpendicular/Hypotenuse, cos θ = Base/Hypotenuse, tan θ = Perpendicular/Base
• Memory code: SOH-CAH-TOA
• sin values for 0°, 30°, 45°, 60°, 90° → 0, 1/2, 1/√2, √3/2, 1
• cos runs the same list backwards: 1, √3/2, 1/√2, 1/2, 0
• tan = sin/cos → 0, 1/√3, 1, √3, not defined

🧮 Key Formulas

sin θ = P/H, cos θ = B/H, tan θ = P/B

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sin row (0°→90°): √0/2, √1/2, √2/2, √3/2, √4/2

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tan 45° = 1, tan 30° = 1/√3, tan 60° = √3

✏️ Easy Example

Q. In a right triangle, perpendicular = 3, base = 4, hypotenuse = 5. Find sin θ and tan θ.

1. sin θ = P/H = 3/5
2. tan θ = P/B = 3/4

Answer: sin θ = 3/5, tan θ = 3/4

🇮🇳 Real-Life Example

A park slide (फिसलपट्टी): its steepness is exactly tan θ — height gained divided by ground distance covered.

📝 Exam-Level Example

Q. Evaluate: sin 30° + cos 60°.

1. sin 30° = 1/2
2. cos 60° = 1/2
3. 1/2 + 1/2

Answer: 1

📝 Exam-Level Example

Q. Find the value of tan 30° × tan 60°.

1. tan 30° = 1/√3, tan 60° = √3
2. (1/√3) × √3

Answer: 1

🪄 Memory Trick

Rebuild the whole table from one line: sin = √0/2 to √4/2 across 0°–90°; reverse it for cos; divide for tan.

⚠️ Common Mistakes

• ❌ Swapping sin and cos values of 30° and 60°
• ❌ Writing tan 90° = 1 — it is not defined

🏆 Exam Tips

• ✅ Write the table on your rough sheet before the paper starts
• ✅ sin and cos never exceed 1 — sanity-check every answer with this

📌 Summary

• SOH-CAH-TOA defines the three ratios
• sin row: √0/2 … √4/2; cos is its reverse
• tan = sin/cos; tan 45° = 1
• sin, cos ≤ 1 always