All Formulas Of Trigonometry Dlass 10

Trigonometry is one of the highest-weightage chapters in the Class 10 Maths board exam. Below is a complete, exam-ready list of every formula you need — trigonometric ratios, identities, complementary angle rules, the standard angle table, and applications in heights and distances.

1. Trigonometric Ratios (of an acute angle in a right triangle)

For a right triangle with angle θ, hypotenuse (H), side opposite to θ (Perpendicular, P), and side adjacent to θ (Base, B):

• sin θ = Perpendicular / Hypotenuse = P/H
• cos θ = Base / Hypotenuse = B/H
• tan θ = Perpendicular / Base = P/B
• cosec θ = Hypotenuse / Perpendicular = H/P
• sec θ = Hypotenuse / Base = H/B
• cot θ = Base / Perpendicular = B/P

Reciprocal Relations

• sin θ = 1 / cosec θ and cosec θ = 1 / sin θ
• cos θ = 1 / sec θ and sec θ = 1 / cos θ
• tan θ = 1 / cot θ and cot θ = 1 / tan θ

Quotient Relations

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

2. Trigonometric Ratios of Standard Angles

Angle →	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	Not defined
cosec θ	Not defined	2	√2	2/√3	1
sec θ	1	2/√3	√2	2	Not defined
cot θ	Not defined	√3	1	1/√3	0

Tip: Learn sin θ values as 0, 1/2, 1/√2, √3/2, 1 (increasing pattern), and cos θ is the same list reversed.

3. Trigonometric Identities (Very Important)

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Rearranged forms often used in problems

• sin²θ = 1 − cos²θ
• cos²θ = 1 − sin²θ
• sec²θ − tan²θ = 1
• cosec²θ − cot²θ = 1

4. Trigonometric Ratios of Complementary Angles

Two angles are complementary if they add up to 90°.

• 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

5. Sign/Behavior Notes (for 0° ≤ θ ≤ 90°)

• sin θ increases from 0 to 1 as θ goes from 0° to 90°.
• cos θ decreases from 1 to 0 as θ goes from 0° to 90°.
• tan θ increases from 0 to undefined (∞) as θ goes from 0° to 90°.

6. Heights and Distances — Key Terms & Formulas

This is the main application of trigonometry in Class 10.

• Line of Sight: The line drawn from the eye of the observer to the point being viewed.
• Angle of Elevation: The angle formed by the line of sight with the horizontal, when the object is above the horizontal level.
• Angle of Depression: The angle formed by the line of sight with the horizontal, when the object is below the horizontal level.

Common working formula

If height = h, distance from observer = d, and angle = θ:

• tan θ = h / d (most commonly used relation in height–distance problems)
• h = d × tan θ
• d = h / tan θ

7. Quick Revision Chart

Formula Type	Formula
Basic ratio	sin θ = P/H, cos θ = B/H, tan θ = P/B
Reciprocal	cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
Quotient	tan θ = sin θ/cos θ, cot θ = cos θ/sin θ
Pythagorean Identity	sin²θ + cos²θ = 1
Pythagorean Identity	1 + tan²θ = sec²θ
Pythagorean Identity	1 + cot²θ = cosec²θ
Complementary	sin(90°−θ) = cos θ, tan(90°−θ) = cot θ, sec(90°−θ) = cosec θ
Heights & Distances	tan θ = height / distance

Tips to Score Full Marks in Trigonometry (Class 10)

1. Memorize the standard angle table (0°, 30°, 45°, 60°, 90°) perfectly — most direct questions come from here.
2. Practice identity-based questions daily; they’re the most repeated question type in boards.
3. For heights and distances, always draw a clean diagram first — mark angle of elevation/depression correctly before applying formulas.
4. Revise complementary angle formulas separately, since students often mix them up with basic ratios.

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