# Sin 2x Formula

Sin 2x formula is one of the double angle formulas. Using this formula, we can find the sine of the angle whose value is doubled. We are familiar that sin is one of the primary trigonometric ratios that is defined as the ratio of the length of the opposite side (of the angle) to that of the length of the hypotenuse in a right-angled triangle. There are various formulas related to sin 2x and can be verified by using basic trigonometric formulas. As the range of sin function is [-1, 1], the range of sin2x is also [-1, 1].

## What Are Sin 2x Formulas?

The sin 2x formula is the double angle identity used for sine in trigonometry. Trigonometry is a branch of mathematics where we study the relationship between the angles and sides of a right-angled triangle. There are two basic formulas for sin 2x:

**sin(2x) = 2 sin x cos x**(in terms of sin and cos)**sin(2x) = (2tan x)/(1 + tan**(in terms of tan)^{2}x)

These are the main formulas of sin 2x. But we can write this formula in terms of sin x (or) cos x alone using the trigonometric identity sin^{2}x + cos^{2}x = 1. They are

- sin 2x = 2 √(1 - cos
^{2}x) cos x (sin 2x formula in terms of cos) - sin 2x = 2 sin x √(1 - sin
^{2}x) (sin 2x formula in terms of sin)

## Derivation of Sin 2x Formula

To derive the formula for sin2x, the sum formula of sin can be used. Let us see the derivation of sin2x step by step:

The sum formula of sin is sin(A + B) = sin A cos B + sin B cos A

Substitute A = B = x in the above equation,

sin(x + x) = sin x cos x + sin x cos x

sin(2x) = 2 sin x cos x

Hence proved.

## Sin 2x Formula in Terms of Tan

We can write the formula of sin 2x in terms of tan or tangent function only. For this, let us start with the sin 2x formula.

sin 2x = 2 sin x cos x

Multiply and divide by cos x. Then

sin 2x = (2 sin x cos^{2}x)/(cos x)

= 2 (sin x/cosx ) · (cos^{2}x)

We know that sin x/cos x = tan x and cos x = 1/(sec x). So

sin 2x = 2 tan x · (1/sec^{2}x)

Using one of the Pythagorean trigonometric identities, sec^{2}x = 1 + tan^{2}x. Substituting this,

sin 2x = (2tan x)/(1 + tan^{2}x)

Therefore, the sin 2x formula in terms of tan is,

**sin 2x = (2tan x)/(1 + tan**^{2}x)

## Solved Examples on sin 2x Formula

**Example 1: **If cos A = 3/5 where A is in quadrant I, then find the value of sin 2A.

**Solution:**

We have

sin^{2}A + cos^{2}A = 1

sin^{2}A = 1- cos^{2}A

sin A = ±√(1 − cos^{2}A)

sin A = ±√(1 − (3/5)^{2})

sin A = ±√(16/25)

sin A = ± 4/5

Since A is in quadrant I, sin A is positive. Thus,

sin A = 4/5

From sin 2x formula, sin 2x = 2 sin x cos x. From this,

sin 2A = 2 sin A cos A

= 2 (4/5) (3/5)

= 24/25

**Answer: **The value of sin 2A = 24/25.

**Example 2: **If sin A = 2/3 where A is in quadrant I, then find the value of sin 2A.

**Solution:**

We have

sin^{2}A + cos^{2}A = 1

cos^{2}A = 1 − sin^{2}A

cos A = ±√(1 − sin^{2}A)

cos A = ±√(1 − (2/3)^{2})

cos A = ±√(5/9)

cos A = ±√5/3

Since A is in quadrant I, cos A is positive. Thus,

cos A = √5/3

Using the sin 2x formula,

sin 2A = 2 sin A cos A

= 2 (2/3) (√5/3)

= 4√5/9

**Answer: **The value of sin 2A = 4√5/9.

**Example 3: **If tan A = 4/3, find the value of sin 2A.

**Solution:**

We have

sin 2x = 2tan(x)/(1 + tan^{2}(x))

So sin 2A = 2tan(A)/(1 + tan^{2}(A))

Putting the value of tan A = 4/3, we get

Sin 2A = 2(4/3)/(1 + (4/3)^{2})

Sin2A = (8/3)/(25/9)

Sin 2A = 24/25

**Answer: **sin 2A = 24/25.

## FAQs on Sin2x Formula

### What is Sin 2x Formula?

Sin 2x formula is the double angle formula of sin function and sin 2x = 2 sin x cos x is the most frequently used formula. But sin 2x in terms of tan is sin 2x = 2tan(x)/(1 + tan^{2}(x)).

### What is the Period of Sin2x?

The period of sin bx is (2π)/b in general. So the period of sin 2x is (2π)/2=π.

### What is Sin 2A in Terms of Cos?

The general formula of sin 2A is, sin 2A = 2 sin A cos A. Using sin^{2}A + cos^{2}A = 1, we get sin A =√(1 - cos^{2}A). Substituting this in the above formula, sin 2A = 2 √(1 - cos^{2}A) cos A. This formula is in the terms of cos or cosine function only.

### How to Prove Sin 2x Formula?

From the sum formula of sin, we have sin (A + B) = sin A cos B + cos A sin B. Substituting A = B = x here, we get, sin 2x = 2 sin x cos x.

### What is Sin 2A in Terms of Sin?

The general formula of sin 2A is, sin 2A = 2 sin A cos A. Using sin^{2}A + cos^{2}A = 1, we get cos A =√(1 - sin^{2}A). Substituting this in the above formula, sin 2A = 2 sin A √(1 - sin^{2}A). This formula is in the terms of sin or sine function only.

### Is Sin 2x Equal to 2 Sin x?

No, sin 2x is not equal to 2 sin x. Infact, sin 2x = 2 sin x cos x from the double angle formula of sin.

### What is Sin 2x in Terms of Tan?

The general formula of sin 2x is sin 2x = 2 sin x cos x = 2 (sin x cos^{2}x)/(cos x) = 2 (sin x/cos x) (1/sec^{2}x) = (2 tan x)/(1 + tan^{2}x). This is sin 2x in terms of tan.