What does the Double Angle Formula Calculator do?

The double angle formula calculator is a great tool if you'd like to see the step by step solutions of the sine, cosine and tangent of double a given angle. Use the calculator above for instant results in your browser.

Is the Double Angle Formula Calculator free to use?

Yes. All Try To Calculator tools are free and do not require an account.

Are my inputs stored or sent to a server?

No. Calculations run locally in your browser. We do not collect the numbers you enter or the results shown.

Can I use the Double Angle Formula Calculator for professional decisions?

This tool is for education and quick estimates. For medical, legal, tax, or financial decisions, verify results with a qualified professional.

Where can I find related calculators?

Browse more Math tools on Try To Calculator at /math, or use the related calculators section on this page.

Double Angle Formula Calculator

Last updated: May 24, 202543 people find this calculator helpful

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Our double angle formula calculator is useful if you'd like to find all of the basic double angle identities in one place, and calculate them quickly. Such identities are useful for proving, simplifying, and solving more complicated trigonometric problems, so it's crucial that you understand and remember them. Don't worry; we'll give you a hand with that!

Whether you're searching for the sin double angle formula, or you'd love to know the derivation of the cos double angles formula, we've got you covered. Keep reading this double angle calculator, and — hopefully — trigonometric identities for double angles won't be your pain in the neck anymore.

What is a double angle? Double angles identities

In this section, we'll show you the double angle identities for sine, cosine, and tangent. To avoid misunderstandings, let's clarify at the beginning what a double angle is:

Double angle means that we increase the given angle by two

So, for example:

$90°$ 90° is the double angle for $45°$ 45°; and
If your given angle equals $\frac{- π}{3}$ 3−π, then the double angle is $\frac{- 2 π}{3}$ 3−2π

🙋 Need a refresh on the measurement units of angles? Our angle converter is here to help!

Now we can proceed with the basic double angles identities:

1. Sin double angle formula

To calculate the sine of a double angle ( $2 θ$ 2θ) in terms of the original angle ( $θ$ θ), use the formula:

sin (2 \cdot θ) = 2 \cdot sin (θ) \cdot cos (θ)

sin(2⋅θ)=2⋅sin(θ)⋅cos(θ)

You can derive this formula from the angle sum identity. As the sum of two sines is:

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

sin(x+y)=sin(x)⋅cos(y)+cos(x)⋅sin(y)

for a double angle we can write it as:

sin (2 \cdot θ) = sin (θ + θ) = sin (θ) \cdot cos (θ) + cos (θ) \cdot sin (θ)

sin(2⋅θ)=sin(θ+θ)=sin(θ)⋅cos(θ)+cos(θ)⋅sin(θ)

Which gives us:

sin (2 \cdot θ) = 2 \cdot sin (θ) \cdot cos (θ)

sin(2⋅θ)=2⋅sin(θ)⋅cos(θ)

2. Cos double angle formula

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

cos (2 \cdot θ) = cos^{2} (θ) - sin^{2} (θ)

cos(2⋅θ)=cos2(θ)−sin2(θ)

Which involves both the sine and the cosine:

cos (2 \cdot θ) = 2 \cdot cos^{2} (θ) - 1

cos(2⋅θ)=2⋅cos2(θ)−1

Or its alternative with the sine:

cos (2 \cdot θ) = 1 - 2 \cdot sin^{2} (θ)

cos(2⋅θ)=1−2⋅sin2(θ)

Analogically to the sine double angles identities, you can derive the first equation from the angle sum and difference identities:

cos (x + y) = cos (x) \cdot sin (y) - sin (y) \cdot sin (y)

cos(x+y)=cos(x)⋅sin(y)−sin(y)⋅sin(y)

For a double angle, it can be expressed as:

cos (2 \cdot θ) = cos (θ + θ) = cos (θ) \cdot cos (θ) - sin (θ) \cdot sin (θ)

cos(2⋅θ)=cos(θ+θ)=cos(θ)⋅cos(θ)−sin(θ)⋅sin(θ)

therefore:

cos (2 \cdot θ) = cos^{2} (θ) - sin^{2} (θ)

cos(2⋅θ)=cos2(θ)−sin2(θ)

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

sin^{2} (θ) + cos^{2} (θ) = 1

sin2(θ)+cos2(θ)=1

🙋 This identity is straightforward if you consider the sine and cosine of an angle as the catheti of a right triangle built on the circle with radius 1 (which, in turn, is the hypotenuse). Calculating the Pythagorean theorem, we can easily find the missing values.

Replace $sin^{2} (θ)$ sin2(θ) with $1 -$ cos2(θ)1−cos2(θ) to get the second equation:

cos (2 \cdot θ) = cos^{2} (θ) - sin^{2} (θ) = cos^{2} (θ) - (1 - cos^{2} (θ)) = 2 \cdot cos^{2} (θ) - 1

cos(2⋅θ)=cos2(θ)−sin2(θ)=cos2(θ)−(1−cos2(θ))=2⋅cos2(θ)−1

Replace $cos^{2} (θ)$ cos2(θ) by $1 -$ sin2(θ)1−sin2(θ) to get the third formula:

cos (2 \cdot θ) = cos^{2} (θ) - sin^{2} (θ) = (1 - sin^{2} (θ)) - sin^{2} (θ) = 1 - 2 \cdot sin^{2} (θ)

cos(2⋅θ)=cos2(θ)−sin2(θ)=(1−sin2(θ))−sin2(θ)=1−2⋅sin2(θ)

3. Tan double angle formula

The formula we use to calculate the tangent of a double angle looks as follows:

tan (2 \cdot θ) = \frac{2 \cdot tan ( θ )}{1 - tan ^{2} ( θ )}

tan(2⋅θ)=1−tan2(θ)2⋅tan(θ)

Similarly, use the sum of tangents formula:

tan (x + y) = \frac{tan ( x ) + tan ( y )}{1 - tan ( y ) \cdot tan ( x )}

tan(x+y)=1−tan(y)⋅tan(x)tan(x)+tan(y)

For a double angle, the equation is then:

tan (2 \cdot θ) = tan (θ + θ) = \frac{tan ( θ ) + tan ( θ )}{1 - tan ( θ ) \cdot tan ( θ )}

tan(2⋅θ)=tan(θ+θ)=1−tan(θ)⋅tan(θ)tan(θ)+tan(θ)

At last, we've found the final equation:

tan (2 \cdot θ) = \frac{2 \cdot tan ( θ )}{1 - tan ^{2} ( θ )}

tan(2⋅θ)=1−tan2(θ)2⋅tan(θ)

Double angles formula calculator - how to use

After all of that, are you wondering how to use this double angle formula calculator? The recommendation is simple — play with it! It won't explode, we promise.

However, if that advice isn't sufficient, here's a short set of instructions:

Decide on the units of the angle. For example, assume we would like to find out what's the double angle sine for an angle with π radians. Choose that unit from the drop-down list (π rad).
Input the angle's value. Let's use $\frac{π}{12}$ 12π as an example. In that case, input $1/12$ 1/12 into the angle box.
Decide whether you'd like to see a step-by-step solution or not. By default, it's shown, but you can hide it at any time.
That's it. The double angle formula calculator has already done the job and found the double angles of sine, cosine, and tangent. For $θ =$ 12πθ=12π, the double angle trigonometric functions look like this:
- $sin (2 \cdot$ θ)=21sin(2⋅θ)=21;
- $cos (2 \cdot$ θ)=23cos(2⋅θ)=23; and
- $tan (2 \cdot$ θ)=33tan(2⋅θ)=33.

Additionally, the double angle formula calculator shows the equivalent of the chosen angle in degrees: $\frac{π}{12} =$ 15°12π=15°

However, if you don't care much about the step-by-step solutions, you can simply use our trigonometric functions calculator — just input double the angle you're interested in directly (so for the example above, enter $\frac{π}{6}$ 6π or $30°$ 30°).

FAQs

What does the Double Angle Formula Calculator do?: The double angle formula calculator is a great tool if you'd like to see the step by step solutions of the sine, cosine and tangent of double a given angle. Use the calculator above for instant results in your browser.
Is the Double Angle Formula Calculator free to use?: Yes. All Try To Calculator tools are free and do not require an account.
Are my inputs stored or sent to a server?: No. Calculations run locally in your browser. We do not collect the numbers you enter or the results shown.
Can I use the Double Angle Formula Calculator for professional decisions?: This tool is for education and quick estimates. For medical, legal, tax, or financial decisions, verify results with a qualified professional.
Where can I find related calculators?: Browse more Math tools on Try To Calculator at /math, or use the related calculators section on this page.

Double Angle Formula Calculator

What is a double angle? Double angles identities

Double angles formula calculator - how to use

FAQs

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