## Double Angle Formula

Trigonometry is an amusing and fundamental branch of Mathematics. We study a number of formulas, theorems and equations in trigonometry which has extensive use in science. In this article, we’ll see a part of this broad area which includes sin function, double angle formula and more specifically double angle formula for sin function. We’ll see its derivation, example and uses of sin2x all formulas.

In trigonometry, one will be able to find many double angle formulas, the “sin 2x” formula is one of them. Using this formula, we can enable ourselves to calculate the sine of the angle whose value is doubled. Sin function is one of the primary trigonometric ratios that is defined as the ratio between the length of the opposite side to that of the length of the hypotenuse in a right-angled triangle.

There are many formulas related to sin 2x that can be derived using basic trigonometric formulas.

Formulas and identities of sin 2x, cos 2x, tan 2x, cot 2x, sec 2x and cosec 2x are known as double angle formulas because they have angle double of the angle present in their formulas.

Sin 2x formula is 2sinxcosx.

Sin 2x =2 sinx cosx

### Derivation of Sin2x Formula

Before going into the actual proof, first, let us take a look at the formula itself.

Sin 2x = 2 sinx cosx

Observe that the sin2x formula is a product of sinx and cosx. We will start by using the known formula in which sin and cos are multiples of each other. This approach leads to a formula that we know as the angle sum formula of sin.

Sin(a+b) = Sin a Cos b + Cos a Sin b

where a and b are angles.

We can replace a and b both as x which gives us,

Sin(x+x) = Sin x Cos x + Cos x Sin x

which can be written as,

Sin(2x)= 2Sin x Cos x

Hence Proved.

### Use of Sin2x All Formula

These double angle formulas and to be more precise cos 2x and sin 2x formulas are used in the large problems of integration and differentiation. Apart from pure mathematics, they are also used in real-life problems of height and distance. The simplification of big problems will make it easier for us to solve them. This simplification is done by double angle formulas of cos 2x and sin 2x formula.

### Examples Based on sin2x Formula

Question: Find the Value 2sinx sin2x Formula in Terms of Cos.

Answer: We can simplify the given expression by substituting the value of sin 2x. We know that

Sin (2x) = 2Sin x Cos x

On substituting the value we get,

2sin x sin2x =2sinx 2sinxcosx

2sinxsin2x =4sin^{2}xcosx

Since we need to get this expression in cos we can use the identity Sin^{2}θ + Cos^{2}θ = 1. We get,

2sinxsin2x = 4(1-cos^{2}x)cosx

2sinxsin2x = 4cosx-4cos^{3}x

Question: Find the Value of sin90^{o}. Use the Double Angle Formula For that.

Answer: We know the double angle formula of sin which is the formula of sin2x as 2sinxcosx

In order to find the value of sin90o, we have to use this formula. We can do so by finding the correct value of x.

2x=90^{o}

\[\Rightarrow x=\frac{90^{0}}{2}\]

x=45^{o}

Now we have got the value of x. Let’s substitute this value into the sin2x formula.

sin(2 x 45^{o}) = 2sin45^{o} cos45^{o}

We know that \[Sin 45^{o}=\frac{1}{\sqrt{2}}\] and \[Cos 45^{o}=\frac{1}{\sqrt{2}}\] Using these values we get,

\[\Rightarrow Sin90^{o}=2\times\frac{1}{\sqrt{2}}\times\frac{1}{\sqrt{2}}\]

\[\Rightarrow Sin90^{o}=2\times\frac{1}{2}\]

sin90^{o} = 1

Hence the required value of sin90^{o} is 1.

Question: Calculate the value of the given expressions sin75^{∘}sin15^{∘}.

Answer: The given expression,

Sin750 sin15o

= Sin ( 90o−15o) sin15o

= cos15o sin15o [as cosx=sin (90∘ − x)]

Multiplying and dividing by 2

= \[\frac {(2 cos15^0 sin15^0)} {2}\]

Now, applying double angle formula of sin2x = 2 sinx cosx

= \[\frac {(sin 30^0)} {2}\]

As sin 30o = \[\frac {1} {2}\]

So, = \[\frac {1} {2}\]*\[\frac {1} {2}\] = \[\frac {1} {4}\] \[\frac {1} {2}\]

Thus the solution for sin75^{o}sin15^{o} will be \[\frac {1} {4}\].

## FAQs on Sin 2x Formula

**1. What is the 2sinx sin2x Formula?**

Here, we have to find the value of 2 sin x sin 2x. We know that sin2x=2sinxcosx.

Using the above formula in the given expression we will get,

2sinx . 2sinxcosx = 4sin2xcosx

And as sin2x = 1 - cos2x, the expression further becomes

⇒ 4 (1 - cos2 x) cos x = 4( cosx - cos3 x)

= 4 cosx - 4 cos3 x .

Thus the expression becomes as

2 sinx sin2x = 4 cos x - 4 cos3x . Hence this is the required result.

2. What is the cos 2x Formula? Prove it.

As proving the sin 2x formula, we can easily prove the cos 2x formula. As know the angle sum formula of cos is

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

Where a and b are angled.

Assuming that both a and b are equal and are equal the x.

Now replace a and b from x. We’ll get

cos(x+x)=cosx cosx - sinx sinx

Which gives us

cos2x=cos2x-sin2x

This is the required cos2x formula.

**3. Prove that the value of sin90 is equal to 1 with the help of the sin 2x formula.**

We know that the value of sin90^{o} is 1 unit. i.e. sin90^{o} = 1.

And in order to prove this statement true, let us first assume the 2x = 90o and from here, We got x = 45^{0}.

Substituting the value in the sin 2x formula, we will be able to prove the given statement.

sin 2x = 2 × sinx × cosx.

sin(2 x 45^{o}) = 2 × sin45^{o} × cos45^{o} ,

We know that the value of these two terms are as sin45^{o} = 12 and cos45^{o} = 12, substituting these values in the equation above.

sin90^{o} = 2 ×12 ×12

sin90^{o} = 2 ×12

sin90^{o} = 1, hence proven the value of sin90^{o} is equal to 1 (with the help of sin 2x formula)

**4. Define sin 2x formula in terms of tan?**

The formula of sin 2x can be written in terms of only tan (tangent function). Let us derive the sin 2x formula with only the tangent formula.

The original formula sin 2x = 2 sinx cosx

Now, dividing and multiplying the equation with cos x

sin 2x = \[\frac {(2 sinx cos2x)} {cosx}\] Or

sin 2x = \[\frac {2(sinx)}{cosx} \times ({cos2x}) \]

And as \[(\frac {sinx} {cosx})\] = tan x and cos x = \[\frac {1} {secx}\] , substituting these values in the equation above we get,

sin 2x = 2 tanx × \[(\frac {1} {sec2x})\]

Using a trigonometric identity, sec2x = 1 + tan2x, the formula will be.

sin 2x = \[\frac {2(tanx)} {(1+tan2x)}\]

Hence, the formula of sin 2x in terms of just tangent functions (tan).

**5. Provide an easy derivation of the sin 2x formula?**

The sin value for the double angle is in the double the value of a product of sin and cos values of a single angle, i.e. sin 2x = 2 sinx cosx. This formula can easily be derived by using the addition formula for sin angles.

The additional formula for sin is given as:

sin(x+y) = sin(x) × cos(y) + cos(x) × sin(y),

where x and y are typically two different angles, but if we assume the x = y, then the equation becomes as follows

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

sin 2x = 2 sin x cos x.