📐 MATHS · CLASS 10

Trigonometry

SOH-CAH-TOA, the unit circle and all identities — visually explained

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📐 The Right Triangle & SOH-CAH-TOA

Trigonometry is the study of relationships between angles and sides in a right triangle.

The 3 Basic Ratios sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent

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On a unit circle, the coordinates of any point are exactly (cos θ, sin θ). Watch the angle sweep!

Point on unit circle: (cos θ, sin θ) | sin²θ + cos²θ = 1

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Memorise these — they appear in almost every trig problem!

Angle (°)	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	∞

💡 Trick for sin: √0/2, √1/2, √2/2, √3/2, √4/2 = 0, ½, 1/√2, √3/2, 1

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These identities are tools — learn them, and any trig problem becomes solvable.

Pythagoreansin²θ + cos²θ = 1

Derived1 + tan²θ = sec²θ

Derived1 + cot²θ = cosec²θ

Ratiotan θ = sin θ / cos θ

Complementarysin(90°−θ) = cos θ

Complementarytan(90°−θ) = cot θ

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Q: Prove that sin²A + cos²A = 1 using a right triangle with hypotenuse c, opposite a, adjacent b.

sin A = a/c cos A = b/c

sin²A + cos²A = a²/c² + b²/c² = (a² + b²)/c²

a² + b² = c² (Pythagoras)

sin²A + cos²A = c²/c² = 1 ✓ Proved!

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Trigonometry conquered! 🎉