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Double Angle Formula Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: May 26, 2024


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\degree is the double angle for 45°45\degree; and
  • If your given angle equals π3\frac {-\pi}{3}, then the double angle is 2π3\frac{-2\pi}{3}

🙋 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 (2θ2\theta) in terms of the original angle (θ\theta), use the formula:

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

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

sin(x ⁣+ ⁣y) ⁣= ⁣sin(x) ⁣ ⁣cos(y) ⁣+ ⁣cos(x) ⁣ ⁣sin(y)\begin{split} \sin(x\!+\!y)\!&=\!\sin(x)\!\cdot\!\cos(y)\!\\ &+\!\cos(x)\!\cdot\!\sin(y) \end{split}

for a double angle we can write it as:

sin(2θ)=sin(θ+θ)=sin(θ) ⁣ ⁣cos(θ)+cos(θ) ⁣ ⁣sin(θ)\begin{split} \sin(2\cdot\theta)&= \sin(\theta+\theta)\\ &=\sin(\theta)\!\cdot\!\cos(\theta)\\ &+\cos(\theta)\!\cdot\!\sin(\theta) \end{split}

Which gives us:

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

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(2θ)=cos2(θ)sin2(θ)\cos(2\cdot\theta)=\cos^2(\theta)-\sin^2(\theta)

Which involves both the sine and the cosine:

cos(2θ)=2cos2(θ)1\cos(2\cdot\theta)=2\cdot\cos^2(\theta)-1

Or its alternative with the sine:

cos(2θ)=12sin2(θ)\cos(2\cdot\theta)=1-2\cdot\sin^2(\theta)

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) ⁣ ⁣sin(y)sin(y) ⁣ ⁣sin(y)\begin{split} \cos(x+y)&=\cos(x)\!\cdot\!\sin(y)\\ &-\sin(y)\!\cdot\!\sin(y) \end{split}

For a double angle, it can be expressed as:

cos(2θ)=cos(θ+θ)=cos(θ) ⁣ ⁣cos(θ)sin(θ) ⁣ ⁣sin(θ)\begin{split} \cos(2\cdot \theta)&=\cos(\theta+\theta)\\ &=\cos(\theta)\!\cdot\!\cos(\theta)\\ &-\sin(\theta)\!\cdot\!\sin(\theta) \end{split}

therefore:

cos(2θ)=cos2(θ)sin2(θ)\cos(2\cdot\theta) = \cos^2(\theta)-\sin^2(\theta)

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

sin2(θ)+cos2(θ)=1\sin^2(\theta)+\cos^2(\theta)=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 sin2(θ)\sin^2(\theta) with 1cos2(θ)1 - \cos^2(\theta) to get the second equation:
cos(2 ⁣ ⁣θ) ⁣= ⁣cos2(θ) ⁣ ⁣sin2(θ) ⁣= ⁣cos2(θ) ⁣ ⁣(1 ⁣ ⁣cos2(θ)) ⁣= ⁣2 ⁣ ⁣cos2(θ)1\begin{split} \cos(2\!\cdot\! \theta) &\!=\! \cos^2(\theta)\!-\!\sin^2(\theta)\\ &\!=\!\cos^2(\theta)\!-\!(1\!-\!\cos^2(\theta))\\ &\!=\!2\!\cdot\!\cos^2(\theta)-1 \end{split}
  • Replace cos2(θ)\cos^2(\theta) by 1sin2(θ)1 - \sin^2(\theta) to get the third formula:
cos(2 ⁣θ) ⁣= ⁣cos2(θ) ⁣ ⁣sin2(θ) ⁣= ⁣(1 ⁣ ⁣sin2(θ)) ⁣ ⁣sin2(θ) ⁣= ⁣1 ⁣ ⁣2 ⁣ ⁣sin2(θ)\begin{split} \cos(2\cdot\!\theta) &\!=\!\cos^2(\theta)\!-\!\sin^2(\theta)\\ &\!=\!(1\!-\!\sin^2(\theta))\!-\!\sin^2(\theta)\\ &\!=\!1\!-\!2\!\cdot\!\sin^2(\theta) \end{split}

3. Tan double angle formula

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

tan(2θ)=2tan(θ)1tan2(θ)\tan(2\cdot\theta)=\frac{2\cdot\tan(\theta)}{1-\tan^2(\theta)}

Similarly, use the sum of tangents formula:

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

For a double angle, the equation is then:

tan(2 ⁣ ⁣θ) ⁣= ⁣tan(θ ⁣+ ⁣θ) ⁣= ⁣tan(θ) ⁣+ ⁣tan(θ)1 ⁣ ⁣tan(θ) ⁣ ⁣tan(θ)\begin{split} \tan(2\!\cdot\!\theta)&\!=\!\tan(\theta\!+\!\theta)\\ &\!=\!\frac{\tan(\theta)\!+\!\tan(\theta)}{1\!-\!\tan(\theta)\!\cdot\!\tan(\theta)} \end{split}

At last, we've found the final equation:

tan(2θ)=2tan(θ)1tan2(θ)\tan(2\cdot\theta)=\frac{2\cdot\tan(\theta)}{1-\tan^2(\theta)}

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:

  1. 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).

  2. Input the angle's value. Let's use π12\frac{\pi}{12} as an example. In that case, input 1/121/12 into the angle box.

  3. 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.

  4. 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=π12\theta = \frac{\pi}{12}, the double angle trigonometric functions look like this:

    • sin(2θ)=12\sin(2\cdot\theta) = \frac{1}{2};

    • cos(2θ)=32\cos(2\cdot\theta) = \frac{\sqrt{3}}{2}; and

    • tan(2θ)=33\tan(2\cdot\theta) = \frac{\sqrt{3}}{3}.

Additionally, the double angle formula calculator shows the equivalent of the chosen angle in degrees: π12=15°\frac{\pi}{12} = 15\degree

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 π6\frac{\pi}{6} or 30°30\degree).

Hanna Pamuła, PhD
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