Sin 2x 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 =4sin²xcosx

Since we need to get this expression in cos we can use the identity Sin²θ + Cos²θ = 1. We get,

2sinxsin2x = 4(1-cos²x)cosx

2sinxsin2x = 4cosx-4cos³x

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^osin15^o will be \[\frac {1} {4}\].