Trigonometry – Complete Revision Notes

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1 Basic Trigonometric Ratios

In a right-angled triangle with angle θ:

Ratio	Definition	Formula
sin θ	Opposite / Hypotenuse	sin θ = P / H
cos θ	Adjacent / Hypotenuse	cos θ = B / H
tan θ	Opposite / Adjacent	tan θ = P / B
cot θ	1 / tan θ	cot θ = B / P
sec θ	1 / cos θ	sec θ = H / B
cosec θ	1 / sin θ	cosec θ = H / P

Remember:
👉 P = Perpendicular, B = Base, H = Hypotenuse

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2 Reciprocal Identities

$$\sin θ = \frac{1}{\csc θ}, \quad \cos θ = \frac{1}{\sec θ}, \quad \tan θ = \frac{1}{\cot θ}$$ $$\csc θ = \frac{1}{\sin θ}, \quad \sec θ = \frac{1}{\cos θ}, \quad \cot θ = \frac{1}{\tan θ}$$

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3 Quotient Identities

$$\tan θ = \frac{\sin θ}{\cos θ}, \quad \cot θ = \frac{\cos θ}{\sin θ}$$

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4 Pythagorean Identities

$$\sin^2 θ + \cos^2 θ = 1$$
$$1 + \tan^2 θ = \sec^2 θ$$
$$1 + \cot^2 θ = \csc^2 θ$$

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5 Trigonometric Values for Standard Angles

θ	sin θ	cos θ	tan θ	cot θ	sec θ	cosec θ
0°	0	1	0	∞	1	∞
30°	1/2	√3/2	1/√3	√3	2/√3	2
45°	1/√2	1/√2	1	1	√2	√2
60°	√3/2	1/2	√3	1/√3	2	2/√3
90°	1	0	∞	0	∞	1

Memory Trick:
👉 sin = √(0/4), √(1/4), √(2/4), √(3/4), √(4/4) for 0°, 30°, 45°, 60°, 90° respectively.

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6 Relation Between Angles

Relation	Formula
Complementary Angles	sin(90° − θ) = cos θ, cos(90° − θ) = sin θ
	tan(90° − θ) = cot θ, cot(90° − θ) = tan θ
	sec(90° − θ) = cosec θ, cosec(90° − θ) = sec θ

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7 Signs of Trig Ratios (Quadrants)

Quadrant	sin	cos	tan
I (0°–90°)	+	+	+
II (90°–180°)	+	−	−
III (180°–270°)	−	−	+
IV (270°–360°)	−	+	−

Trick:
👉 All Students Take Calculus
A – All positive
S – Sine positive
T – Tan positive
C – Cos positive

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8 Trigonometric Ratios of (180° ± θ), (360° − θ)

Angle Form	sin	cos	tan
sin(180° − θ)	+sin θ
sin(180° + θ)	−sin θ
cos(180° − θ)	−cos θ
cos(180° + θ)	−cos θ
tan(180° − θ)	−tan θ
tan(180° + θ)	+tan θ

(Same for other multiples of 90° with sign changes based on quadrant.)

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9 Sum and Difference Formulas

Formula Type	Formula
sin(A ± B)	sinA cosB ± cosA sinB
cos(A ± B)	cosA cosB ∓ sinA sinB
tan(A ± B)	(tanA ± tanB) / (1 ∓ tanA tanB)

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10 Double Angle Formulas

$$\sin 2A = 2\sin A \cos A$$
$$\cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A = 2\cos^2 A - 1$$
$$\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$$

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11 Half Angle Formulas

$$\sin^2 \frac{A}{2} = \frac{1 - \cos A}{2}$$ $$\cos^2 \frac{A}{2} = \frac{1 + \cos A}{2}$$ $$\tan \frac{A}{2} = \frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}$$

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12 Product to Sum and Sum to Product

Formula	Equivalent Form
sinA sinB	½[cos(A−B) − cos(A+B)]
cosA cosB	½[cos(A−B) + cos(A+B)]
sinA cosB	½[sin(A+B) + sin(A−B)]

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13 Trigonometric Equations

Commonly used results:

• sin θ = sin α → θ = nπ + (−1)^n α

• cos θ = cos α → θ = 2nπ ± α

• tan θ = tan α → θ = nπ + α

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14 Height and Distance Basics

Formulas:

$$\tan θ = \frac{\text{Height}}{\text{Distance}}$$ $$\cot θ = \frac{\text{Distance}}{\text{Height}}$$

Example:

If a tower 50 m high casts a shadow of 50√3 m,
tan θ = 50 / (50√3) = 1/√3 → θ = 30°

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✅  Quick Memory Sheet

Category	Key Formula
Reciprocal	sinθ = 1/cosecθ, cosθ = 1/secθ
Pythagorean	sin²θ + cos²θ = 1
tan-cot	tanθ = sinθ/cosθ
Double Angle	sin2A = 2sinAcosA
Half Angle	tan(A/2) = sinA/(1+cosA)
Complementary	sin(90°−θ) = cosθ
Heights & Distances	tanθ = height/base

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💡 Tips for Exams

✅ Always convert given angles into standard forms (0°–90°).
✅ Remember ASTC sign rule.
✅ Simplify using sin² + cos² = 1 wherever possible.
✅ For quick questions, visualize triangle ratios.
✅ Practice standard values (30°, 45°, 60°).