Sin A Plus B Formula

Sin A Plus B or sin(A + B) is a common formula in trigonometry used to find various values of sine. The sine of the sum of two angles A and B (often denoted as sin (A + B)) can be expressed using the sine and cosine of the individual angles A and B. In this article, we will discuss formula in detail.

What are Trigonometric Identities?

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables within their respective domains. These identities are fundamental tools in trigonometry and are used to simplify expressions, solve equations, and prove other mathematical statements.

Some common trigonometric identities include:

• Reciprocal Identities
• Quotient Identities
• Co-Function Identities
• Pythagorean Identities
• Sum and Difference Identities

In this article, we will the sin (a + b) identity in detail including it’s proof.

What is Sin(a + b)?

Sin (a+b) is an important formula used to calculate the sine value of the angle formed by the sum of the given two angles. It is a standard trigonometric identity used to find the sine trigonometric function for the sum of angles. It is generally used to simplify the calculations.

Formula for Sin(a + b)

Let us see the formula used for calculating the sine value of a compound angle ‘a+b’.

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

Proof of Sin(a + b) Formula

Assumption: Assume that there are two acute angles a and b and the sum of a and b is also an acute angle meaning ‘a+b'<90^o

Construction: Draw a line OL and generate another line at an angle ‘a’ from the original line (call it line OQ). This line is further rotated in the same direction making an angle ‘b’ with the initial drawn line(call it line OP). Note that the angle ‘SOP’ in the given figure is acute. Now draw two perpendiculars from point P and Q on line OL.

This construction is given in the following figure:

Proof:

From the right triangle PTQ we get,

∠TPQ = 90° – ∠PQT = ∠TQO = alternate ∠QOL = a.

In triangle QOS,

sin a = QS/OQ

In triangle POQ

sin b = PQ/OP

cos b = OQ/OP

In triangle PTQ,

cos a = PT/PQ

Now, from the right-angled triangle PRO, we get,

sin (a + b) = PR/OP

⇒ sin (a + b) = (PT + TR)/OP

⇒ sin (a + b) = PT/OP + TR/OP

⇒ sin (a + b) = PT/OP + QS/OP

⇒ sin (a + b) = PT/PQ ∙ PQ/OP + QS/OQ ∙ OQ/OP

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

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

How to Apply Sin(a + b) Formula?

To apply we need to know the two angles sum of whose sine of compound angle needs to be found. Let us consider a=45^o and b=45^o to understand this formula.

• Write down the compound angle formula as sin (a + b) = sin a cos b + cos a sin b.
• Now, you need to put in the values of a and b.
• Then identify the values of a and b and put these values in the formula

Let’s consider an example.

Example: Find the value of sin 90 using sin(a + b) formula.

Solution:

As we know, sin 90 = sin (45 + 45) = sin (30 + 60)

Sin(45+45) = sin45.cos45 +cos45.sin45

⇒ Sin (90) = 1/√2 . 1/√2 + 1/√2 . 1/√2

⇒ Sin (90) = 1/2 + 1/2

⇒ Sin (90) = 1

Some Other Similar Identities

Some other similar identities includes:

• 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
• tan (A + B) = (tan A + tan B)/(1 – tan A tan B)
• tan (A – B) = (tan A – tan B)/(1 + tan A tan B)

Read More,

Trigonometric Identities	Inverse Trigonometric Identities
Sum and Difference Formulas	Double Angle Identities
Half Angle Formulas	Triple Angle Identities

Solved Examples on Sin (a + b)

Example 1: Find the value of sin(75) using the compound angle formula.

Solution:

From the compound angle formula, sin (a + b) = sin a cos b + cos a sin b

∴ sin(75)=sin(30+45)

∴ a = 300 and b = 450

∴ sin(75)=sin(30).cos(45)+cos(30).sin(45)

∴ sin(75)= 1/2. 1/√2 +√3/2. 1/√2

∴ sin(75)= (1+√3)/2√2

Example 2: Find the value of sin(105) using the compound angle formula.

Solution:

From the compound angle formula sin (a + b) = sin a cos b + cos a sin b

∴ sin(105)=sin(60+45)

∴ a = 600 and b = 450

∴ sin(105) = sin(60).cos(45)+cos(60).sin(45)

∴ sin(105) = √3/2. 1/√2+1/2. 1/√2

∴ sin(105) = (1+√3)/2√2

Practice Questions on Sin (a + b)

Q1: Find the value of sin(85) using the compound angle formula.

Q2: Find the value of sin(20)cos(25) + sin(25)cos(20).

Q3: Prove that sin (a+b) is not the same as sina + sinb.

FAQs on Sin (a + b) Formula

Define sine.

The sine function, denoted by sin (θ), is a fundamental trigonometric function defined for an angle θ in a right-angled triangle.

What are compound angle formulas?

Compound angle formulas, also known as angle addition formulas, are trigonometric identities that express trigonometric functions of the sum or difference of two angles in terms of trigonometric functions of the individual angles.

What is the formula for cos A minus B?

Formula for cos A minus B is given as: cos (A – B) = cos A cos B + sin A sin B

Write formula for sin A minus B.

Formula for sin A minus B is: sin (A – B) = sin A cos B – cos A sin B