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Sin(a + b) is one of the important trigonometric identities used in trigonometry. We use the sin(a + b) identity to lớn find the value of the sine trigonometric function for the sum of angles. The expansion of sin (a + b) helps in representing the sine of a compound angle in terms of sine & cosine trigonometric functions. Let us understand the sin(a+b) identity & its proof in detail in the following sections.

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1.What is Sin(a + b) Identity in Trigonometry?
2.Sin(a + b) Compound Angle Formula
3.Proof of Sin(a + b) Formula
4.How to Apply Sin(a + b)?
5.FAQs on Sin(a + b)

What is Sin(a + b) Identity in Trigonometry?


Sin(a + b) is the trigonometry identity for compound angles. It is applied when the angle for which the value of the sine function is to be calculated is given in the khung of the sum of angles. The angle (a + b) represents the compound angle.

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Sin(a + b) Compound Angle Formula


Sin(a + b) formula is generally referred lớn as the addition formula in trigonometry. The sin(a + b) formula for the compound angle (a + b) can be given as,

sin (a + b) = sin a cos b + cos a sin b


Proof of Sin(a + b) Formula


The proof of expansion of sin(a + b) formula can be done geometrically. Let us see the stepwise derivation of the formula for the sine trigonometric function of the sum of two angles. In the geometrical proof of sin(a + b) formula, let us initially assume that 'a', 'b', and (a + b) are positive acute angles, such that (a + b) = (PT + TQ)/OP= PT/OP + TQ/OP= PT/OP + RS/OP= PT/PR ∙ PR/OP + RS/OR ∙ OR/OP= cos (∠TPR) sin b + sin a cos b= sin a cos b + cos a sin b, (since we know, ∠TPR = a)

Therefore, sin (a + b) = sin a cos b + cos a sin b.


How to lớn Apply Sin(a + b)?


The expansion of sin(a + b) can be used khổng lồ find the value of the sine trigonometric function for angles that can be represented as the sum of standard angles in trigonometry. We can follow the steps given below to lớn learn to apply sin(a + b) identity. Let us evaluate sin(30º + 60º) to lớn understand this better.

Step 1: Compare the sin(a + b) expression with the given expression lớn identify the angles 'a' và 'b'. Here, a = 30º & b = 60º.Step 2: We know, sin (a + b) = sin a cos b + cos a sin b.⇒ sin(30º + 60º) = sin 30ºcos 60º + sin 60ºcos 30ºSince, sin 60º = √3/2 , sin 30º = 1/2, cos 60º = 1/2, cos 30º = √3/2⇒ sin(30º + 60º) = (1/2)(1/2) + (√3/2)(√3/2) = 1/4 + 3/4 = 1Also, we know that sin 90º = 1. Therefore the result is verified.

Related Topics on Sin(a+b):

Here are some topics that you might be interested in while reading about sin(a+b).

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Let us have a look a few solved examples to lớn understand sin(a + b) formula better.