Sin2x Formula: Derivation, Solved Examples & Uses

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Sin2x Formula is a double-angle formula in Trigonometry that is used to find the sine of the angle with a double value. Sin2x, Cos2x, and Tan2x are the double-angle formulas used in Trigonometry.

Sine is a primary trigonometric ratio which is the ratio of the opposite side to that of the hypotenuse in a right-angled triangle.
It is generally denoted by the abbreviation sin.
The range of the Sine Function is [-1, 1], and the range of Sin2x is also [-1, 1].
Sin2x Formula is used to solve different trigonometric, integration, and differentiation problems.

Sin2x Formula is given as

Sin 2x = 2Sin X Cosx

Table of Content

What is Sin2x?
Sin2x Formula
Derivation of Sin2x Formula
Sin2x Formula in Terms of Tan
Sin²x Formula
Sin2x Formula Solved Examples
Uses of Sin2x Formula
Things to Remember
Previous Years’ Questions
Sample Questions

Key Terms: Sin2x, Sin2x Formula, Sine Function, Trigonometric Ratios, Double Angle Formula, Tangent Function, Cosine Function

What is Sin2x?

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Sin2x is a trigonometric formula in trigonometry that is used to simplify the various trigonometric expressions involving double angles.

Sin2x is twice the product of the sine function and cosine function which is given as sin2x = 2sin x cosx.
It can also be expressed in terms of Tangent Function as well.
It is used in various problems related to Trigonometry, Integration, and Differentiation.
It is called the double-angle formula as it involves double angles trigonometric functions, i.e. sin 2x.

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Relevant Concepts
Cos2x Formula	2 Cos A Cos B Formula	Sin Cos Formulas
Sin 180 Degrees	Tan2x Function	Sin Cos Tan

Sin2x Formula

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Sin2x Formula is a double-angle formula used for Sine Function in Trigonometry. It is an integral trigonometric identity that is used for a wide range of trigonometric and integration problems. There are two major basic Sin2x Formulas:

Sin2x Formula In Terms of Sin and Cos

Sin2x Formula In terms of Sin and Cos is given as follows:

sin2x = 2 sin x cos x

Sin2x Formula In Terms of Tan

Sin2x Formula In terms of Tan is given as follows:

sin2x = (2tan x)/(1 + tan²x)

Sin2x Formula can be written in terms of sin x (or) cos x alone using the trigonometric identity sin²x + cos²x = 1. It can be written as sinx = √(1 - cos²x) and cosx = √(1 - sin²x) using the trigonometric identity. Thus, Sin2x Formulas in terms of cos and sin are:

Sin2x Formula in Terms of cos: sin2x = 2 √(1 - cos²x) cos x
Sin2x Formula in Terms of sin: sin2x = 2 sin x √(1 - sin²x)

Solved Example

Example: Find the value of sin2A if cos A = 3/5 where A is in quadrant I.

Solution: Using the Pythagorean Identity, sin²A + cos²A = 1.

sin²A = 1 - cos²A
sin A = ±√(1 − cos²A)
sin A = ±√(1 − (3/5)²)
sin A = ±√(16/25)
sin A = ± 4/5

As A is in quadrant I, sin⁡ A is positive. Therefore,

sin A = 4/5

Using Sin2x Formula, sin2x = 2 sin x cos x. Thus,

sin2A = 2 sin A cos A

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

= 24/25

Thus, the value of sin⁡2A is 24/25.

Derivation of Sin2x Formula

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Sin2x Formula can be derived by using the Sum Angle Formula for the Sine Function. Using Trigonometric Identities, the Sum Formula of Sin is

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

Substitute x = y in the formula,

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

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

sin2x = 2 sin x cos x

Thus, the Sin2x Formula is derived.

Trigonometric Functions Detailed Video Explanation

Sin2x Formula in Terms of Tan

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Sin 2x can also be written in terms of the Tangent Function only.

Sin2x = 2 sin x cos x

On multiplying and dividing the above equation by cos x, we get

sin2x = (2 sin x cos²x)/(cos x)

= 2 (sin x/cosx ) × (cos²x)

As, sin x/cos x = tan x and cos x = 1/(sec x), thus,

sin2x = 2 tan x × (1/sec²x)

Using Pythagorean Trigonometric Identities, sec²x = 1 + tan²x.

Substituting this, we get

sin2x = (2tan x)/(1 + tan²x)

Thus, sin2x formula in terms of tan is sin2x = (2tan x)/(1 + tan²x).

Sin²x Formula

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Sin Square x or Sin²x is a mathematical formula derived from the Pythagorean Identities and double angle formulas of the cosine function. It is used to solve complex integration problems and to prove different trigonometric identities. In order to derive sin²x Formula, trigonometric identities sin²x + cos²x = 1 and the double angle formula of the cosine function cos 2x = 1 – 2 sin²x are used. Sin²x can be expressed in terms of cos² x and cos2x using these identities.

Sin²x Formula in Terms of Cos x

Using Trigonometric Identities, sin²x + cos²x = 1 and subtracting cos²x from both the sides of the Identity, we get

sin²x + cos²x -cos²x = 1 - cos²x

It implies that

sin²x = 1 - cos²x

Thus, the Sin²x Formula using Pythagorean Identity is sin²x = 1 - cos²x. It is used to simplify trigonometric expressions.

Sin²x Formula in Terms of Cos 2x

Using the double-angle formula, cos 2x = 1 – 2sin²x and interchanging the terms, it can be written as

2 sin²x = 1 - cos2x ⇒ sin²x = (1 - cos2x)/2

Thus, the formula of sine square x using the cos2x formula is

sin²x = (1 - cos2x)/2

This formula is used to solve complex integration problems.

Thus, the two basic formulas of sin²x are:

sin²x = 1 - cos²x ⇒ sin²x = 1 - cos²x
sin²x = (1 - cos2x)/2 ⇒ sin²x = (1 - cos2x)/2

Sin2x Formula Solved Examples

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Here are some solved examples on Sin2x Formula for a better understanding of the formula:

Example 1: Determine the value of 2sinx sin2x in terms of Cos.

Solution: Sin2x Formula is given as Sin (2x) = 2Sin x Cos x.

Simplify the given expression by substituting the value of Sin2x.

2sinx sin2x = 2sinx 2sinx cosx

2sinx sin2x = 4sin²xcosx

As we need to get this expression in Cos, use the identity Sin²θ + Cos²θ = 1.

We get,

2sinx sin2x = 4(1-cos²x)cosx

2sinx sin2x = 4cosx – 4cos³x

Example 2: Find the value of sin 90 Degrees. Use the Sin2x Formula.

Solution: Sin2x Formula is given as Sin (2x) = 2Sin x Cos x. To find the value of sin 90 degrees, Sin2x Formula needs to be used.

2x = 90^o

x = 90°/2

x = 45°

The value of x is obtained. Substituting its value into the Sin2x Formula,

Sin (2 x 45°) = 2sin45° cos45°

We know that sin45° = 1/√2 and cos 45° = 1/√2. Using these values we get,

Sin 90°=2×1/√2 x 1/√2

Sin 90° = 1

Thus, the required value of sin 90° is 1.

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Tan 0 Degrees	Product to Sum Formulas	Integration of Trigonometric Functions
Sin 120	Tangent 3 Theta Formula	Differentiation Rules

Uses of Sin2x Formula

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Sin2x Formula is a double-angle formula in Trigonometry.

Double angle formulas especially cos2x and sin2x Formulas are used in problems of Integration and Differentiation.
It is also used in real-life problems of height and distance.
Sin2x Formula also helps in the simplification of problems with tedious calculations.
It helps in the simplification of various complex trigonometric expressions.

Things to Remember

Trigonometry is a branch of Mathematics that studies the relationship between the sides and angles of a triangle.
Sine is a trigonometric ratio which is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
Sin2x Formula is a double-angle formula in trigonometry used to find the sine of the angle whose value is doubled.
Sin2x Formula in terms of Cos and Sin is sin2x = 2 sin x cos x.
Sin2x Formula in terms of Tan is sin2x = (2tan x)/(1 + tan²x).
There are six double-angle formulas in Trigonometry namely Sin2x, Cos2x, Tan2x, Cot2x, Sec2x, and Cosec2x.

Previous Years’ Questions

The greatest and least value of sin x cos x are…
The number of solutions for the equation Sin 2x + Cos… (KCET - 2008)
The number of solutions of the equation sin2x + 2sinx… (UPSEE - 2018)
If the angles of a triangle are in the ratio 4 :1 :1, then… (JEE Advanced - 2003)
Which one of the following is not true… (AMUEEE - 2013)
If the angles A,B, and C of a triangle are in arithmetic… (JEE Advanced - 2010)
The principal value of sin… (JEE Advanced - 1986)
If sin(θ+α) = cos(θ+α), then the… (COMEDK UGET - 2007)
A tower subtends an angle of 30^∘ at a point on the same level…
Find the maximum and minimum values of…

Sample Questions

Ques. Find the value of Sin2x if sin x = 3/5. (3 Marks)

Ans. Given that, sin x = 3/5.

Thus, cos x = 4/5.

Using the Sin2x Formula, we get

Sin2x = 2 sin x cos x

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

= 24/25

Thus, the value of Sin2x if sin x = 3/5 is 24/25.

Ques. Find the value of Sin2x if cosec x = 17/8. (3 Marks)

Ans. Given that, cosec x = 17/8.

Thus, sin x = 8/17 and cos x = 15/17.

Using the Sin2x Formula, we get

Sin2x = 2 sin x cos x

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

= 240/289

Thus, the value of Sin2x if cosec x = 17/8 is 240/289.

Ques. Find the value of Sin 75 Sin 15. (3 Marks)

Ans. According to the given question,

Sin 75 Sin 15 = Sin (90-15) Sin 15

= Cos 15 Sin 15 …. (cos x = sin (90-x))

= 1/2 sin 30 (Sin 2x = 2sin x cosx)

= ½ x 1/2 (As sin 30 = 1/2 )

= 1/4

Ques. If cos x = 12/13, what will be the value of Sin2x? (3 Marks)

Ans. It is given that cos x = 12/13.

Thus, sin x = 5/13.

Using the Sin2x Formula, we get

Sin2x = 2 sin x cos x

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

= 120/169

Thus, the value of Sin2x if cos x = 12/13 is 120/169.

Ques. If tan x = 12/5, what will be the value of Sin2x? (3 Marks)

Ans. Given that, tan x = 12/5.

Using the Sin2x Formula, we get

sin2x = (2tan x)/(1 + tan²x)

= 2 × (12/5) / {1 + (12/5)²}

= 120/169

Thus, the value of Sin2x if tan x = 12/5 is 120/169.

Ques. Explain Cos2x Formula. (3 Marks)

Ans: Cos2x Formula is a double-angle trigonometric formula. It can be derived from the Angle Sum Formula of Cos. The angle sum formula of cos is:

Cos (a + b) = cos a. cos b – sin a. sin b

Now replace a and b with x to get:

Cos (x + x) = cos x cos x- sin x sin x

Cos2x = cos2x – sin2x

Thus, the Cos2x Formula is Cos2x = Cos2x – Sin2x

Ques. What is Trigonometry? (3 Marks)

Ans. Trigonometry is the study of relationships between the angles and sides of a right-angled- triangle. It was discovered by the Greek Mathematician named Hipparchus.

It helps to find the missing or unknown angles or sides of a right triangle using trigonometric formulas, functions, or trigonometric identities.
The angles can be either measured in degrees or radians in Trigonometry.
The most commonly used trigonometric angles for calculations are 0°, 30°, 45°, 60° and 90°.

Ques. If cot x = 15/8, what will be the value of Sin2x? (3 Marks)

Ans. It is given that cot x = 15/8

tan x = 1 / cot x = 1 / (15/8) = 8 / 15

Using the Sin2x Formula, we get

sin2x = (2tan x)/(1 + tan²x)

= 2 × (18 / 15) / {1 + (18 / 15)²}

= 240/289

Thus, the value of Sin2x if cot x = 15/8 is 240/289.

Ques. If cosec x = 13/12, what will be the value of Sin2x? (3 Marks)

Ans. Given that cosec x = 13/12.

Using Pythagoras Theorem, sin x = 12/13 and cos x = 5/13.

Using the Sin2x Formula, we get

Sin2x = 2 sin x cos x

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

= 120/169

Thus, the value of Sin2x if cosec x = 13/12 is 120/169.

Ques. Find the value of Sin2x if sec x = 5/3. (3 Marks)

Ans. Given that, sec x = 5/3.

Using Pythagoras Theorem, cos x = 3/5 and sin x = 4/5.

Using the Sin2x Formula, we get

Sin2x = 2 sin x cos x

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

= 24/25

Thus, the value of Sin2x if sec x = 5/3 is 24/25.

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Sin2x Formula: Derivation, Solved Examples & Uses

What is Sin2x?