The important trigonometrical ratios of compound angle formulae are given below:
sin (A + B) = sin A cos B + cos A sin B
sin (A - B) = sin A cos B - cos A sin B
cos (A + B) = cos A cos B - sin A sin B
cos (A - B) = cos A cos B + sin A sin B
sin ( A + B) sin (A - B) = sin^2 A – sin^2 B = cos^2 B – cos^2 A
cos ( A + B) cos (A - B) = cos^2 A – sin^2 B = cos^2 B – sin^2 A
tan (A + B) = tan A + tan B/1 - tan A tan B
tan (A - B) = tan A - tan B/1 + tan A tan B
cot (A + B) = cot A cot B - 1/cot B + cot A
cot (A - B) = cot A + cot B + 1/cot B - cot A
tan (A + B + C) = tan A + tan B + tan C - tan A tan B tan C/1- tan A tan B - tan B tan C - tan C tan A.
Now we will learn how to use the above formulae for solving different types of trigonometric problems on compound angle.
1. Using the formula of cos (A - B) = cos A cos B + sin A sin B prove that cos (π/2 - x) = sin x, for all real numbers x.
Proof:
cos (π/2 - x) = cos π/2 cos x + sin π/2 sin x, [Applying the formula of cos (A - B) = cos A cos B + sin A sin B]
= 0 × cos x + 1 × sin x, [Since we know that cos π/2 = 0 and sin π/2 = 1]
= 0 + sin x
= sin x Proved
Therefore cos (π/2 - x) = sin x.
2. Using the formula of cos (A + B) = cos A cos B - sin A sin B prove that cos (π/2 + x) = - sin x, for all real numbers x.
Proof:
cos (π/2 + x) = cos π/2 cos x - sin π/2 sin x, [Applying the formula of cos (A + B) = cos A cos B - sin A sin B]
= 0 × cos x - 1 × sin x, [Since we know that cos π/2 = 0 and sin π/2 = 1]
= 0 - sin x
= - sin x Proved
Therefore cos (π/2 + x) = - sin x
3. Using the formula of sin (A + B) = sin A cos B + cos A sin B prove that sin (π/2 + x) = cos x, for all real numbers x.
Proof:
sin (π/2 + x) = sin π/2 cos x + cos π/2 sin x, [Applying the formula of sin (A + B) = sin A cos B + cos A sin B]
= 1 × cos x + 0 × sin x, [Since we know that sin π/2 = 1 and cos π/2 = 0]
= cos x + 0
= cos x Proved
Therefore sin (π/2 + x) = cos x
4. Using the formula of sin (A - B) = sin A cos B - cos A sin B prove that sin (π/2 - x) = cos x, for all real numbers x.
Proof:
sin (π/2 - x) = sin π/2 cos x - cos π/2 sin x, [Applying the formula of sin (A - B) = sin A cos B - cos A sin B]
= 1 × cos x - 0 × sin x, [Since we know that sin π/2 = 1 and cos π/2 = 0]
= cos x - 0
= cos x Proved
Therefore sin (π/2 - x) = cos x
11 and 12 Grade Math
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