2 Sin a Cos a Formula

2 sin a cos a is a trigonometric formula that is equal to the sine of angle 2a, i.e., it is given by 2 sin a cos a = sin 2a. It is one of the important trigonometric identities that is used to solve various trigonometric and integral problems. 2 sin a cos a formula is also called the double angle formula of sine function as it is equal to sin 2a, where 2a is twice the angle a. This formula can also be expressed in terms of tan a.

Let us explore the 2 sin a cos a formula, derive the formula using the sin (a + b) formula, and understand its application to solve different mathematical problems. We will also some a few examples using 2 sin a cos a formula for a better understanding of its application.

2 sin a cos a formula is an important trigonometric formula that is equal to sin 2a. Mathematically, it is written as sin 2a = 2 sin a cos a. It can also be expressed in terms of tan a as well. The two ways in which 2 sin a cos a formula can be written are:

• 2 sin a cos a = sin 2a
• 2 sin a cos a = (2 tan a)/(1 + tan²a)

The first form of this formula is the most commonly used form and it is used to simplify complex trigonometric functions and solving problems. We can also express 2 sin a cos a formula using sin²a + cos²a = 1 formula as:

• 2 sin a cos a = 2 √(1 - cos²a) cos a
• 2 sin a cos a = 2 sin a √(1 - sin²a)

Now, we will prove the 2 sin a cos a formula using the angle sum formula of the sine function, i.e., sin (a + b) = sin a cos b + sin b cos a. Assume a = b in this formula and let us derive the 2 sin a cos a formula step-wise. We have,

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

⇒ sin (a + a) = sin a cos a + sin a cos a [Assuming a = b]

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

⇒ 2 sin a cos a = sin (2a)

Hence, we have proved that 2 sin a cos a is equal to sin 2a.

Next, we will derive the formula of 2 sin a cos a in terms of tan a. We have derived that 2 sin a cos a = sin (2a). Now, if we multiply and divide 2 sin a cos a by cos a, then we have

2 sin a cos a = (2 sin a cos a) × (cos a)/(cos a)

= 2 (sin a/cos a) (cos²a)

Now, we know that sin x/cos x = tan x and 1/cos x = sec x or 1/sec x = cos x. Therefore, we have

2 sin a cos a = (2 tan a)/(sec²a)

= (2 tan a)/(1 + tan²a) [using trigonometric formula 1 + tan²x = sec²x]

Hence, the 2 sin a cos a formula in terms of tan a is given by, 2 sin a cos a = (2 tan a)/(1 + tan²a)

So far, we have derived the formula for 2 sin a cos a. Next, let us understand the application of these formulas in solving different problems. Let us solve a few examples to learn how to apply 2 sin a cos a formula.

Example 1: Find the value of sin 120° using 2 sin a cos a formula.

Solution: We know the values of trigonometric functions for specific angles. So, we have

sin 120° = sin (2 × 60°)

⇒ sin 120° = 2 sin 60° cos 60° (Because 2 sin a cos a = sin (2a))

⇒ sin 120° = 2 × √3/2 × 1/2

⇒ sin 120° = √3/2

The formula can also be conversely used to find the value of 2 sin a cos a using sin 2a.

Example 2: Determine the value of 2 sin 15° cos 15°.

Solution: As we know the values of sine function for specific angles and 2 sin a cos a = sin (2a), we have

2 sin 15° cos 15° = sin (2 × 15°)

⇒ 2 sin 15° cos 15° = sin 30°

⇒ 2 sin 15° cos 15° = 1/2

Important Notes on 2 sin a cos a

• 2 sin a cos a formula is also commonly known as the sin 2a formula.
• 2 sin a cos a = sin (2a)
• 2 sin a cos a = (2 tan a)/(1 + tan²a)
• 2 sin a cos a = 2 √(1 - cos²a) cos a
• 2 sin a cos a = 2 sin a √(1 - sin²a)

Related Topics on 2 sin a cos a

• Cos a Cos b
• Cos 2x
• Cos 3a

What is 2 Sin a Cos a Formula in Trigonometry?

2 sin a cos a formula is an important trigonometric formula that is equal to sin 2a. Mathematically, it is written as sin 2a = 2 sin a cos a. It can also be expressed in terms of tan a as well.

Are Sin 2a and 2 Sin a Cos a Formulas the Same?

Yes, Sin 2a Formula and 2 Sin a Cos a Formula are the same as sin 2a = 2 sin a cos a. In fact, 2 sin a cos a formula is also commonly known as the sin 2a formula.

How to Derive 2 Sin a Cos a Formula?

2 Sin a Cos a can be derived using the angle sum formula of the sine function sin (a + b) by considering a = b.

What is 2 Sin a Cos a in Terms of Tan a?

The 2 sin a cos a formula in terms of tan a is given by, 2 sin a cos a = (2 tan a)/(1 + tan²a).

What is 2 Sin a Cos a Formula Commonly Known As?

2 sin a cos a formula is also commonly known as the sin 2a formula.