Trigonometric Identities for Class 10th – Formulas

Here, you will learn various trigonometric identities for class 10th and formulas of trigonometry.

Let’s begin-

In a right angle triangle

sin\(\theta\) = \(p\over h\); cos\(\theta\) = \(b\over h\); tan\(\theta\) = \(p\over b\); cosec\(\theta\) = \(h\over p\); sec\(\theta\) = \(h\over b\) and cot\(\theta\) = \(b\over p\)

where ‘p’ is perpendicular ; ‘b’ is base and ‘h’ is hypotenuse.

Signs of Trigonometric functions in different quadrants

trigonometric ratio

Basic Trigonometric Identities for Class 10th :

(1)  \(sin\theta\).\(cosec\theta\) = 1

(2)  \(cos\theta\).\(sec\theta\) = 1

(3)  \(tan\theta\).\(cot\theta\) = 1

(4)  \(tan\theta\) = \(sin\theta\over{cos\theta}\)   \(cot\theta\) = \(cos\theta\over{sin\theta}\)

(5)  \(sin^2\theta\) + \(cos^2\theta\) = 1

(6)  \(sec^2\theta\) – \(tan^2\theta\) = 1

(7)  \(cosec^2\theta\) – \(cot^2\theta\) = 1

Trigonometric Ratios of the sum & difference of two angles :

(1)   sin(A + B) = sin A cos B + cos A sin B

(2)   sin(A – B) = sin A cos B – cos A sin B

(3)   cos(A + B) = cos A cos B – sin A sin B

(4)   cos(A – B) = cos A cos B + sin A sin B

(5)   tan(A + B) = \(tan A + tan B\over {1 – tan A tan B}\)

(6)   tan(A – B) = \(tan A – tan B\over {1 + tan A tan B}\)

(7)   cot(A + B) = \(cot B cot A – 1\over {cot B + cot A}\) 

(8)   cot(A – B) = \(cot B cot A + 1\over {cot B – cot A}\)

Formulae to transform the product into sum or difference :

(i)   2 sin A cos B = sin(A + B) + sin(A – B)

(ii)   2 cos A sin B = sin(A + B) – sin(A – B)

(iii)   2 cos A cos B = cos(A + B) – cos(A – B)

(iv)   2 sin A sin B = cos(A – B) – cos(A + B)

Formulae to transform the sum or difference into product :

(i)    sin C + sin D = 2 sin(\(C + D\over 2\)) cos(\(C – D\over 2\))

(ii)   sin C – sin D = 2 cos(\(C + D\over 2\)) sin(\(C – D\over 2\))

(iii)  cos C + cos D = 2 cos(\(C + D\over 2\)) cos(\(C – D\over 2\))

(iv)  cos C – cos D = 2 sin(\(C + D\over 2\)) sin(\(D – C\over 2\))

Trigonometric ratios of sum of more than two angles :

(i)   sin(A + B + C) = sinAcosBcosC + sinBcosAcosC + sinCcosAcosB – sinAsinBsinC

(ii)  cos(A + B + C) = cosAcosBcosC – sinAsinBcosC – sinAcosBsinC – cosAsinBsinC

(iii)  tan(A + B + C) = \(tanA + tanB + tanC – tanAtanBtanC\over {1 – tanAtanB – tanBtanC – tanCtanA}\)

Trigonometric ratios of mutiple angles :

(i)  sin2A = 2sinAcosA = \(2tanA\over {1+tan^2A}\)

(ii)  cos2A = \(cos^2A\) – \(sin^2A\) = \(2cos^2A\) – 1 = 1 – \(2sin^2A\) = \(1 – tan^2A\over {1 + tan^A}\)

(iii)  1 + cos2A = \(2cos^2A\)

(iv)  1 – cos2A = \(2sin^2A\)

(v)   tanA = \(1 – cosA\over {sin2A}\) = \(sin2A\over {1+cos2A}\)

(vi)  tan2A = \(2tanA\over {1-tan^2A}\)

(vii)  sin3A = 3sinA – \(4sin^3A\)

(viii)  cos3A = \(4cos^3A\) – 3cosA

(ix)  tan3A = \(3tanA – tan^3A\over {1 – 3tan^2A}\)

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