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Sin2x Formula

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  • Last Updated : 10 Nov, 2022
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Sin2x formula is among the very few important formulas of trigonometry used to solve various problems in mathematics. It is among the various double-angle formulas used in trigonometry. This formula is used to find the sine of the angle with a double value. Sin is among the primary trigonometric ratios that are given by taking the ratio of perpendicular to that of the hypotenuse in a right-angled triangle. The range of sin2x is [-1, 1].

Sine ratio is calculated by computing the ratio of the length of the opposing side of an angle divided by the length of the hypotenuse. It is denoted by the abbreviation sin.

Right-angled Triangle

 

If θ is the angle formed between the base and hypotenuse of a right-angled triangle then,

sin θ = Perpendicular/Hypotenuse

What is Sin2x?

Sin2x is a formula used in trigonometry to solve various mathematical, and other problems. It helps to simplify various trigonometric expressions involving double angles. Sin2x is expressed in different forms using various trigonometric functions. The most common formula of sin2x is, sin2x = 2 sinx cosx. It can also be expressed in terms of the tan function.

Sin2x Formula

Sin 2x is a double-angle identity in trigonometry. Because the sin function is the reciprocal of the cosecant function, it may alternatively be written sin2x = 1/cosec 2x. It is an important trigonometric identity that may be used for a wide range of trigonometric and integration problems. The value of sin 2x is repeated every π radians, that is, sin 2x = sin (2x + π). It has a much narrower graph than sin x. It’s a trigonometric function that calculates the sin function of a double angle. Various other trigonometric ratios are used along with this to solve mathematical problems.

sin 2x = 2 sin x cos x

Sin 2x Derivation Formula

The formula for sin 2x can be derived by using the sum angle formula for the sine function.

Using Trigonometric Identities, sin (x + y) = sin x cos y + cos x sin y

In order to find sine for double angle, we must put x = y

Putting x = y we get,

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

sin 2x = sin x cos x + sin x cos x

sin 2x = 2 sin x cos x

This derives the formula for the double angle of the sine ratio.

Sin2x Formula in Terms of Tan

sin 2x can also be given in terms of the tan function. Let’s take a look at how Sin 2x is given in terms of tan x

sin 2x = 2 sin x cos x

Multiplying and dividing it by cos x. 

sin 2x = (2 sin x cos2x)/(cos x)

         = 2 (sin x/cosx ) × (cos2x)               {sin x/cos x = tan x and cos x = 1/(sec x)}

sin 2x = 2 tan x × (1/sec2x)                       {sec2x = 1 + tan2x}

sin 2x = (2tan x)​/(1 + tan2x)

Thus, the sin  2x formula in terms of tan is sin 2x = (2tan x)​/(1 + tan2x).

Sin2x Formula in Terms of Cos

sin 2x can also be given in terms of cos function. Let’s take a look at how Sin 2x is given in terms of cos x

sin 2x = 2 sin x cos x …(1)

we know that sin x = √(1 – cos2x) using this in eq (1)

sin 2x = 2 √(1 – cos2x) × cos x 

This is the required formula for Sin 2x in terms of Cos x.

Sin2x Formula in Terms of Sin

sin 2x can also be given in terms of sin function. Let’s take a look at how Sin 2x is given in terms of sin x

sin 2x = 2 sin x cos x …(1)

we know that cos x = √(1 – sin2x) using this in eq (1)

sin 2x = (2 sin x )× √(1 – sin2x) 

This is the required formula for Sin 2x in terms of Sin x.

Sin2x

Sin2x formulas are used to solve complex mathematical problems, they are also used to simplify trigonometric identities. Two formulas for sin2x can be derived using the Pythagorean Theorem and the double angle formulas of the cosine function.

Sin2x Formula

For the derivation of the sin2x formula, we use the trigonometric identities sin2x + cos2x = 1 and the double angle formula of cosine function cos 2x = 1 – 2 sin2x. Using these identities, sin2x can be expressed in terms of cos2 x and cos2x. Let us derive the formulas:

Sin2x Formula in Terms of Cos x

We know that, using trigonometric identities,

sin2x + cos2x = 1 using the equation and sending cos2x to the left-hand side which changes its sign we get,

sin2x = 1 – cos2x

Sin2x Formula in Terms of Cos 2x

We know that, using the double-angle formula,

cos 2x = 1 – 2sin2x using the equation and separating sin2x to one side we get,

sin2x = (1 – cos 2x) / 2

Therefore, the two basic formulas of sin2x are:

sin2x = 1 – cos2

sin2x = (1 – cos 2x) / 2

Important Formulas

sin 2x = 2 sin x cos x
sin 2x = (2tan x)​/(1 + tan2x)

Other Formulas

sin2x  = 1 – cos2
sin2x = (1 – cos2x)/2

Solved Examples on Sin 2x Formula

Example 1. If sin x = 3/5, find the value of sin 2x using the formula.

Solution:

We have, sin x = 3/5.

Clearly, cos x = 4/5.

Using the formula we get,

sin 2x = 2 sin x cos x

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

= 24/25

Example 2. If cos x = 12/13, find the value of sin 2x using the formula.

Solution:

We have, cos x = 12/13.

Clearly, sin x = 5/13.

Using the formula we get,

sin 2x = 2 sin x cos x

= 2 (5/13) (12/13)

= 120/169

Example 3. If tan x = 12/5, find the value of sin 2x using the formula.

Solution:

We have, tan x = 12/5.

Using the formula we get,

sin2x = (2tan x)​/(1 + tan2x).

         = 2 × (12/5) / {1 + (12/5)2}

         = 120/169

Example 4. If cosec x = 17/8, find the value of sin 2x using the formula.

Solution:

We have, cosec x = 17/8.

Clearly sin x = 8/17 and cos x = 15/17.

Using the formula we get,

sin 2x = 2 sin x cos x

= 2 (8/17) (15/17)

= 240/289

Example 5. If cot x = 15/8, find the value of sin 2x using the formula.

Solution:

We have, cot x = 15/8

tan x = 1 / cot x = 1 / (15/8) 

        = 8 / 15

Using the formula we get,

sin2x = (2tan x)​/(1 + tan2x).

         = 2 × (18 / 15) / {1 + (18 / 15)2}

        = 240/289

Example 6. If cosec x = 13/12, find the value of sin 2x using the formula.

Solution:

We have, cosec x = 13/12.

Clearly sin x = 12/13 and cos x = 5/13 (using pythagoras theorem)

Using the formula we get,

sin 2x = 2 sin x cos x

          = 2 (12/13) (5/13)

         = 120/169

Example 7. If sec x = 5/3, find the value of sin 2x using the formula.

Solution:

We have, sec x = 5/3.

Clearly cos x = 3/5 and sin x = 4/5 (using pythagoras theorem)

Using the formula we get,

sin 2x = 2 sin x cos x

          = 2 (4/5) (3/5)

          = 24/25

FAQs on Sin 2x Formula

Question 1: What is the differentiation of Sin 2x?

Answer:

The differentiation of sin 2x is 2cos 2x

Question 2: What is the integration of Sin2x?

Answer: 

The integration of sin 2x is (-cos 2x) / 2

Question 3: What is the Sin 2x formula in terms of the tan function?

Answer:

Sin 2x formula in terms of the tan function is sin2x = (2tan x)​/(1 + tan2x).

Question 4: Write the formula for tan 2x.

Answer:

The formulas used for tan 2x are:

  • tan2x = 2tan x / (1−tan2x)
  • tan2x = sin 2x/cos 2x

Question 5: Write the formula for cos 2x.

Answer:

The formulas used for cos 2x are:

  • cos2x = cos2x – sin2x
  • cos2x = 2cos2x – 1
  • cos2x = 1 – 2sin2x
  • cos2x = (1 – tan2x)/(1 + tan2x)

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