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Double Angle Calculator

Created by Anna Szczepanek, PhD
Reviewed by Komal Rafay
Last updated: Jan 18, 2024


Omni's double angle calculator is here to introduce you to the trig identities involving double angles. We know they appear repeatedly in your math assignments! We're here to help by explaining how to calculate double angles and simplify various trig expressions with the help of the double angle formulas. Scroll down!

🙋 Not yet familiar with sines and cosines? Before diving into trig identities for double angles, make sure to discover the basics with the help of Omni's trigonometric functions calculator.

Sine double angle formula

Let us start with the formula for the sine of a double angle. Here (and throughout this article), θ\theta stands for an arbitrary angle. The formula reads:

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

As you can see, to compute the sine of two times theta, you need to know both the sine and cosine of theta. What to do if you know only one of them?

Recall that the sine and cosine are linked by the Pythagorean trigonomeric identity:

sin2(θ)+cos2(θ)=1.\small \sin^2(\theta)+\cos^2(\theta)=1.

Hence, whenever you know either the sine or cosine of theta, you can compute the sine of two theta by passing through the Pythagorean trig identity.

Let us move on to the cosine of a double angle.

Cosine double angle formulas

The cosine double angle formula reads

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

however, sometimes you may see a version of it using only sine or only cosine, that is:

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

or

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

You can move between all these three formulas by applying, again, the Pythagorean trig identity sin2(θ)+cos2(θ)=1\small \sin^2(\theta)+\cos^2(\theta)=1.

Tangent double angle formula

Finally, the time has come to discuss the double angle formula for the tangent function:

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

As you can see, it involves only the tangent of the initial angle. Note that the formula is well defined if tan(θ)1\tan(\theta) \neq 1, i.e., if θπ4\theta \neq \frac \pi 4 (i.e., 4545^\circ). This is because the double angle then is π2\frac \pi 2 (9090^\circ), where, as we know, the tangent function is undefined.

How to use this double angle calculator?

To use Omni's double angle calculator efficiently, you just need to pick the trig function (sine, cosine, and tangent) and feed it with an angle, choosing first the units among:

  • degrees;
  • radians; and
  • π × radians.

The last option is the ideal one if your angle is of the form, e.g., π7\frac \pi 7 or 29π\frac 2 9\pi.

Once you enter all the data, the result will appear at once along with an explanation. Enjoy!

Similar Omni tools

Double angle identities are so ubiquitous in assignments and real-life problems (really!) that we've created a whole batch of tools dedicated to these trig identities, each highlighting a slightly different aspect:

FAQ

What is a double angle in trig?

In trigonometry, a double angle is an angle whose measure was doubled, that is, multiplied by 2. For example:

  • 20° doubled is 40°;
  • 1 rad doubled is 2 rad; and
  • π/6 rad doubled is π/3 rad.

Is 2sinx the same as sin2x?

No, 2×sin(x) does not equal sin(2×x). To see why, observe that the values of 2×sin(x) fill in the interval [-2, 2], whereas the values of sin(2×x) fill in [-1, 1]. The underlying reason for this is the fact that sine is not a linear function.

How do I calculate the tangent of double angle?

To determine the tan(2θ) given tan(θ):

  1. Square tan(θ).
  2. Subtract the result from 1, i.e., compute 1 - tan²(θ).
  3. Multiply tan(θ) by 2.
  4. Divide the number from Step 3 by that from Step 2.
  5. That's it! If you struggle with computations, don't hesitate to use an online calculator for double angles.

What is tan(70°) given tan(35°)?

tan(70°) = 2.75. To get this value when we know that tan(35°) = 0.70, we apply the tan double angle formula:

tan(2θ) = 2×tan(θ) / [1 - tan²(θ)].

Plugging in θ = 35°, we obtain

tan(70°) = 2 × 0.70 / [1 - 0.70²] = 2.75, as claimed.

Anna Szczepanek, PhD
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