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Double Angle Calculator

Double Angle Calculator

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An online double angle calculator can help you to determine all basic double angle identities of the given angle. Basic trigonometric formulas are difficult to remember, so, use this online double angle formula calculator for computing all double angle identities such as sin θ, cos θ, and tan θ  with the units in degree, radian, and pi radian.

Let’s begin with a basic definition!

What is Double Angle?

Trigonometric double angle is linked with the three general ratios: sine, cosine, and tangent. These ratios show the relationship among the sides of a right triangle, with respect to particular angles in the triangle.

Therefore, double angle refers to multiply the angle by two or add the same quantity to the original amount. For instance, if you have 40 oranges and we double the amount by adding 40 more oranges or just multiply by 2.

Both of these concepts apply to doubling the angle. Correspondingly, double angle indicates the following:

$$Sin (x + x) = Sin (2x)$$

$$Cos (x + x) = Cos (2x)$$

$$Tan (x + x) = Tan (2x)$$

Double Angle Formula:

If we have an angle θ, and the double angle is 2θ. Then, the formulas which are used by the double angle calculator is:

$$Sine (2θ) = 2sin (θ) cos (θ)$$

$$Cosine (2θ) = 1 – 2sin^2(θ)$$

$$Cosine (2θ) = cos^2(θ) – sin^2(θ)$$

$$Cosine (2θ) = 2cos^2(θ) -1$$

$$Tangent (2θ) = 2 tan(θ) / 1 – tan^2(θ)$$

How to Calculate a Double Angle?

We can calculate the double angle of trigonometric functions by substituting the several values in double angle formulas for 2θ. First of all, you need to know the trigonometric values for all functions, then you can use the above formulas to determine the double angle.

However, you can use a double angle calculator which helps you to solve all basic trigonometric functions.

Example:

Find double angle of sin (60)o.

Solution:

Using the  above formula:

$$sin (60)^o = sin (2. 30^o)$$

$$2 sin(30^o) cos(30^o)$$

$$ \frac{1}{2}. \sqrt{3}/2$$

$$sin (60)^o = \sqrt{3}/2$$

Double Angle Identities:

Now start with the basic double angles identities:

Sine Double Angle Formula:

To calculate the double angle of the sine formula:

$$Sine (2θ) = 2sin (θ) cos (θ)$$

The sine double angle formula

Now, derive the formula from the angle sum identity.

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

For the double angle we can write as:

$$Sin (2θ) = sin (θ + θ) = cos(θ).sin(θ) + sin(θ).cos(θ)$$

That gives the formula which is used by double angle calculator:

$$Sin (2θ) = 2 . sin(θ) . cos(θ)$$

Well, the Online Sine Calculator will compute the sine trigonometric values for the given angle in degree, radian, or the π radians. Also, you can be able to find cosine from the source of calculator-online.

you can use a free online calculator to find the values of cosine.

Cosine Double Angle Formula:

There are few formulas for the cosine double angle. The three popular cos of double angle equations are:

The double angle formula cosine:

$$= cos^2(θ) – sin^2(θ)$$

$$= 2cos^2(θ) -1$$

$$= 1 – 2sin^2(θ)$$

You can derive the equation from the difference identities and angle sum:

$$cos(x + y) = cos(x).cos(y) – sin(y).sin(x)$$

So, it can be expressed as

$$Cos (2θ) = cos(θ + θ) = cos (θ).cos (θ) – sin (θ).sin(θ)$$

therefore

$$cos(2θ) = – sin^2(θ) + cos^2(θ)$$

To find the other two forms, use the Pythagorean trigonometric identity:

$$sin^2θ + cos^2θ = 1$$

Now, replace \(sin^2θ \text{ by} 1 – cos^2θ\) to get the other equation:

$$Cos (2θ) = cos^2(θ) – sin^2(θ)$$

By simplify this

$$cos^2θ – (1 – cos^2(θ))$$

$$cos^2(θ) – 1$$

Then, replace \(cos^2θ by 1 – sin^2θ\) to get another formula:

$$cos(2θ) = cos^2(θ) – sin^2(θ)$$

Here’s the cosine formula, which is used by the double angle calculator:

$$(1 – sin^2(θ)) – sin^2(θ) = 1 – 2 * sin^2(θ)$$

However, the Online Pythagorean Theorem calculator helps to calculate the length of any missing sides of a right triangle.

Tan Double Angle Formula:

The formula for the tan of a double angle is:

The tangent double angle formula

$$Tangent (2θ) = 2 tan(θ) / 1 – tan^2(θ)$$

Similarly, the sum of tangents formula:

$$Tan (x + y) = (tan (y) + tan(x)) / 1 – tan(y).tan(x)$$

then a double angle equation for tangent is:

$$tan(2θ) = tan(θ + θ)$$

Similarly

$$= (tan(θ) + tan(θ)) / 1 – tan(θ) * tan(θ)$$

Now, found the final equation:

$$tan(2θ) = 2 . tanθ / (1 – tan^2θ)$$

How Double Angle Calculator Works?

An online double angle identities calculator will determine all trigonometric function by applying these steps:

Input:

  • First, select the unit in degree, radian, and pi radian as per requirement
  • Then plug in the angle value in the related portion
  • Press the calculate button

Output:

  • The double angle formula calculator displays all trigonometric values with a step-by-step solution.
  • You can do these calculations Nemours time by click on recalculate button.

FAQs:

Why are trigonometric identities important?

Trigonometric equations are always true, and they’re commonly used to solve geometry and trigonometry problems and understand different mathematical properties. Knowing key trigonometric identities helps you understand and remember the important mathematical principles and solve thousands of math problems.

What are the six trigonometric ratios?

There are six trig ratios, sin, cos, tan, CSC, sec, and cot. These are commonly referred to as ratios since they can be determined in form of the sides of a right-angled triangle for a particular angle θ.

Wrap Up:

Use this online double angle calculator to solve the trigonometric functions such as sin 2θ, cos 2θ, and tan 2θ. The trig functions of some specific angles may even seem obvious since we have worked with them many times. So, use this free online calculator to find the 100% accurate values of the unknown trigonometric functions.

Reference:

From the source of Wikipedia: Double-angle, triple-angle, and half-angle formulae, sine, cosine, and tangent of multiple angles.

From the source of Cliffs Notes: Double‐Angle and Half‐Angle Identities, Pythagorean identity, Double Angle Identities.

From the source of Brilliant: Double Angle Formulas, Hyperbolic Double Angle Formulas, Double Angle Identities.