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Sin 2 Theta Calculator

Created by Davide Borchia
Reviewed by Komal Rafay
Last updated: Jan 18, 2024


The sin 2 theta calculator is the perfect help for your math homework and when you need a refresh of this fundamental trigonometric identity. Keep reading this short article to learn:

  • How to calculate sin 2 theta: the fundamental formula;
  • How to calculate sin 2 theta using other trigonometric identities and the tangent;
  • Other tools related to the double angle identities;

and much more!

How to calculate the sin 2 theta formula

The formula for the sine of a double angle is a trigonometric identity, an equation that compares two mathematical expressions and remains valid for any values in a specified range. In particular, the identity for sin 2 theta calculates the value of the sine of an angle, knowing the sine values and cosine of the angle with half the amplitude. To derive the formula for the sine of the double angle, we can start with the formula for the sine of the compound angle. The formula for the sine of a compound angle is:

sin(θ+φ)=sin(θ)cos(φ)+sin(φ)cos(θ)\scriptsize \sin(\theta+\varphi) = \sin(\theta)\cos(\varphi)\!+\!\sin(\varphi)\cos(\theta)

To obtain this formula, we can use a beautiful graphic proof:

Starting from the light blue triangle, we can find the values of the sides of the two smaller triangles with angle θ\theta. Once we collect enough data, we can build the last triangle that, thanks to the property of parallel lines, has an angle with value θ+φ\theta+\varphi.

Once we know how to compute the sine of a compound angle, we can perform a simple substitution: φ=θ\varphi=\theta. This way, the compound angle becomes a double angle:

sin(θ+φ)=sin(θ+θ)=sin(2θ)\scriptsize \sin(\theta+\varphi) = \sin(\theta+\theta) = \sin(2\theta)

By performing the same substitution, we can find the formula to calculate sin 2 theta:

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

Other ways to calculate sin 2 theta

Let's use other trigonometric identities to calculate sin 2 theta in various ways. The first alternative formula uses:

  • The Pythagorean identity: sin2(θ)+cos2(θ)=1\sin^2(\theta)+\cos^2(\theta) = 1; and
  • The formula for the square of a polynomial: (a+b)2=a2+b2+2ab(a+b)^2 = a^2+b^2+2ab.

Start by noticing that the result of the formula we found in the previous section resembles a double product, in particular of the square of the sum of sine and cosine of theta:

(sin(θ)+cos(θ))2=sin2(θ)+cos2(θ)+2sin(θ)cos(θ)\scriptsize \begin{split} \left(\sin(\theta)+\cos(\theta)\right)^2& =\sin^2(\theta) + \cos^2(\theta)\\ &+2\sin(\theta)\cos(\theta) \end{split}

Using the Pythagorean identity, we can rewrite this equality this way:

(sin(θ)+cos(θ))2=1+2sin(θ)cos(θ)\scriptsize \begin{split} \left(\sin(\theta)+\cos(\theta)\right)^2& =1+2\sin(\theta)\cos(\theta) \end{split}

And eventually, rearranging the order of the factors, we find:

sin(2θ)=2sin(θ)cos(θ)=(sin(θ)+cos(θ))21\scriptsize \begin{split} \sin(2\theta) &= 2\sin(\theta)\cos(\theta)\\ &=\left(\sin(\theta)+\cos(\theta)\right)^2 -1 \end{split}

If we introduce the tangent, we can find a third way to calculate sin 2 theta. Start by rewriting the result of the first formula to find the tangent:

2sin(θ)cos(θ)=2tan(θ)cos2(θ)\scriptsize 2\sin(\theta)\cos(\theta) = 2\tan(\theta)\cos^2(\theta)

Move the square of the cosine at the denominator, and use once again the Pythagorean identity:

2tan(θ)cos2(θ)=2tan(θ)sin2(θ)+cos2(θ)cos2(θ)\scriptsize 2\tan(\theta)\cos^2(\theta)=\frac{2\tan(\theta)}{\frac{\sin^2(\theta)+\cos^2(\theta)}{\cos^2(\theta)}}

Split the denominator to find the tangent:

2tan(θ)sin2(θ)+cos2(θ)cos2(θ)=2tan(θ)tan2(θ)+1\scriptsize \frac{2\tan(\theta)}{\frac{\sin^2(\theta)+\cos^2(\theta)}{\cos^2(\theta)}} = \frac{2\tan(\theta)}{\tan^2(\theta)+1}

Tools to calculate the double angle identities

Why stop at the sine? Omni offers you a complete suite of tools to calculate the double angle identities:

FAQ

How do I calculate sin 2 theta?

To calculate the sin 2 theta, use the double angle formula, a particular case of the compound angle formula.

  1. Start with the compound angle formula: sin(α + ß) = sin(α)cos(ß) + sin(ß)cos(α).
  2. Substitute the angle ß with α: sin(α + α) = sin(2α) = 2sin(α)cos(α).

The result is the double angle identity for the sine.

Can I use the double angle formula to calculate sin 4 theta?

Yes, but the formula is not as nice as the one for the double angle. To calculate sin 4 theta, follow these simple steps:

  1. Calculate sin(4Θ) using the double angle identity: sin(4Θ) = 2sin(2Θ)cos(2Θ).
  2. Rewrite the cosine using the Pythagorean identity sin(4Θ) = 2sin(2Θ)sqrt(1 - sin²(2Θ)).
  3. Use again the double angle identity to calculate sin 4 theta as a function of sin(Θ) and cos(Θ): sin(4Θ) = 2(2sin(Θ)cos(Θ))sqrt(1 - (2sin(Θ)cos(Θ))²).

What is the sine of 80° if sin(40°) = 0.6428?

The value of sin(80°) is 0.9848. To calculate it, use the formula for sin 2 theta, following these steps:

  1. Calculate the cosine of 40° using the Pythagorean identity: cos(40°) = sqrt(1 - sin²(40°)) = sqrt(1 - 0.6428^2)) = 0.766.

2.Use the double angle formula: sin(2Θ) = 2sin(Θ)cos(Θ): sin(80°) = 2sin(40°)cos(40°) = 2 · 0.6428 · 0.766 = 0.9848.
That's it! The double angle identities are often used to find the values of the trigonometric functions of uncommon angles.

Davide Borchia
Angle θ
deg
Step by step solution?
yes, please
Double sine
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