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Sin(a + b)
Sin(a + b) is one of the important trigonometric identities used in trigonometry. It is one of sum and difference formulas. It says sin (a + b) = sin a cos b + cos a sin b.
We use the sin(a + b) identity to find the value of the sine trigonometric function for the sum of angles. The expansion of sin a plus b formula helps in representing the sine of a compound angle in terms of sine and cosine trigonometric functions. Let us understand the sin(a+b) identity and its proof in detail in the following sections.
1.  What is Sin(a + b)? 
2.  Proof of Sin(a + b) Formula 
3.  How to Apply Sin(a + b)? 
4.  FAQs on Sin(a + b) 
What is Sin(a + b)?
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 form of the sum of angles. The angle (a + b) represents the compound angle.
Sin(a + b) Compound Angle Formula
Sin(a + b) formula is generally referred to as one of the addition formulas in trigonometry. The sin a plus 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) < 90. But this formula, in general, is true for any positive or negative value of a and b.
To prove: sin (a + b) = sin a cos b + cos a sin b
Construction: Assume a rotating line OX and let us rotate it about O in the anticlockwise direction. OX makes out an acute ∠XOY = a, from starting position to its initial position. Again, the rotating line rotates further in the same direction and starting from the position OY, thus making out an acute angle given as, ∠YOZ = b. ∠XOZ = a + b < 90°.
On the bounding line of the compound angle (a + b) take a point P on OZ, and draw PQ and PR perpendiculars to OX and OY respectively. Again, from R draw perpendiculars RS and RT upon OX and PQ respectively.
Proof: From triangle PTR we get, ∠TPR = 90°  ∠PRT = ∠TRO = alternate ∠ROX = a.
Now, from the rightangled triangle PQO we get,
sin (a + b) = PQ/OP
= (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 Apply Sin(a + b)?
The expansion of sin(a + b) can be used to 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 learn to apply sina plus b identity. Let us evaluate sin(30º + 60º) to understand this better.
 Step 1: Compare the sin(a + b) expression with the given expression to identify the angles 'a' and 'b'. Here, a = 30º and 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 30º
From trig table, 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 = 1
Also, we know that sin 90º = 1. Therefore the result is verified.
Try: Find the value of sin 75º using sin (a + b) formula.
☛Related Topics:
Examples Using Sin(a + b)

Example 1: Find the exact value of sin 165º using expansion of sin(a + b).
Solution:
Since, the values of sine and cosine functions can be easily calculated for 120º and 45º, we can write 165º as (120º + 45º).
⇒sin(165º) = sin(120º + 45º)
= sin120ºcos45º + sin 45ºcos120º
= (√3/2)(1/√2) + (1/√2)(1/2)
= (√3/2√2)  (1/2√2)
= (√3  1)/2√2
= (√6  √2)/4Answer: ∴ sin 165º = (√6  √2)/4.

Example 2: Apply the sin(a + b) formula to find the expansion of the double angle formula sin 2θ.
Solution:
We can write sin 2θ = sin(θ + θ)
Applying sin(a + b) = sin a cos b + cos a sin b
Put a = b = θ in this formula:
sin 2θ = sinθcosθ + sinθcosθ = 2sinθcosθ
Answer: ∴ sin 2θ = 2sinθcosθ 
Example 3: Derive the formula of sin (a  b) using the sin (a + b) formula.
Solution:
We know that
sin(a + b) = sin a cos b + cos a sin b
Replace "b" with "b" everywhere in this formula:
sin (a + (b)) = sin a cos (b) + cos a sin (b)
sin (a  b) = sin a cos b + cos a ( sin b)
sin (a  b) = sin a cos b  cos a sin b
Answer: ∴ sin (a  b) = sin a cos b  cos a sin b
FAQs on Sin (a + b)
What is Sin (a + b) Formula?
Sin (a + b) is one of the important trigonometric identities also called sine addition formula in trigonometry. Sin(a + b) can be given as, sin (a + b) = sin a cos b + sin b cos a, where 'a' and 'b' are angles.
☛Also Check: sin ( a  b) Formula
What is the Formula of Sin (a + b)?
The sin(a + b) formula is used to express the sin compound angle formula in terms of sin and cosine of individual angles. Sin(a + b) formula in trigonometry can be given as, sin (a + b) = sin a cos b + cos a sin b.
How to Prove Sin (a + b) Formula?
The proof of sin(a + b) formula can be given using the geometrical construction method. We initially assume that 'a', 'b', and (a + b) are positive acute angles, such that (a + b) < 90. Click here to understand the stepwise method to derive sin(a + b) formula.
What are the Applications of Sin a plus b Formula?
sin(a + b) can be used to find the value of sine function for angles that can be represented as the sum of standard or simpler angles. Thus, making the deduction easier. It can also be used in finding the expansion of other double and multiple angle formulas.
What is Expansion of Sin (a + b)?
The expansion of sin(a + b) is given as, sin (a + b) = sin a cos b + cos a sin b. Here, a and b are the measures of angles.
How to Find the Value of Sin 15º Using Sin(a + b) Identity.
The value of sin 15º using (a + b) identity can be calculated by first writing it as sin[(45º+(30º)] and then applying sin(a + b) identity.
⇒sin[(45º+(30º)] = sin 45ºcos(30)º + sin(30)ºcos 45º = (1/√2)(√3/2) + (1/2)(1/√2) = (√3/2√2)  (1/2√2) = (√3  1)/2√2 = (√6  √2)/4
How to Find Sin(a + b + c) using Sin (a + b)?
We can express sin(a+b+c) as sin((a+b)+c) and expand using sin(a+b) and cos(a+b) formula as, sin(a+b+c) = sin(a+b).cos c + sin c.cos(a+b) = cos c · (sin a cos b + cos a sin b) + sin c · (cos a cos b  sin a sin b) = sin a cos b cos c + cos a sin b cos c + cos a cos b sin c  sin a sin b sin c.
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