Double Angle Formula

In trigonometry, it is really important to know and able to use identities. One of the main and crucial categories of identities is Double Angle Formula. In this post, we talked about different trig identities and formulas. Double Angle Formulas are trigonometric identities that simplify a trigonometric function of $2x$ as of trigonometric functions of $x$. First we recommend to visit the previous post to review identities. Here in this post, first we recall the Double Angle Formula and then we see some examples.

Double Angle Formulas:

Here we have main Double Angle Formulas.

  • $\sin (2 u) =2 \sin u \cos u$.
  • $\cos (2 u) =\cos ^{2} u-\sin ^{2} u$.
  • $\cos (2 u) =2 \cos ^{2} u-1$.
  • $\cos (2 u) =1-2 \sin ^{2} u$.
  • $\tan (2 u) =\dfrac{2 \tan u}{1-\tan ^{2} u}$.

Examples of Double Angle Formulas

Before checking the solutions, try to solve it by yourself first.

Example 1: If $\sin x=\dfrac{3}{5}$ and $x$ is in the first quadrant, find $\tan (2 x)$.

Solution: Since $x$ is in the first quadrant, both $\sin x$ and $\cos x$ are positive. Now we use the Pythagorean identity $\sin^{2} x+\cos^{2} x=1$ to find $\cos x$. From this identity we have, $\dfrac{9}{25} +\cos^{2} x=1$, so $\cos x=\dfrac{4}{5}$. Then $\tan x=\dfrac{\sin x}{\cos x}=\dfrac{\dfrac{3}{5}}{\dfrac{4}{5}}=\dfrac{3}{4}$. Now by the last Double Angle Formula, we have $\tan (2 u)=\dfrac{2 \dfrac{3}{4}}{1-\dfrac{9}{16} }=\dfrac{\dfrac{3}{2}}{\dfrac{7}{16}}=\dfrac{48}{14}=\dfrac{24}{7}$.

Example 2: Prove the following identity $\dfrac{1-\cos 2x}{1+\cos 2x}=\tan^2 x$.

Solution: To solve this problem we use the double angle formula for $\cos 2x$. We simplify the left side to get the right side. By the double angle formula we have $\dfrac{1-\cos 2x}{1+\cos 2x}=\dfrac{1-(1-2 \sin ^{2} x)}{1+2 \cos ^{2} x-1}=\dfrac{\sin ^{2} x}{\cos ^{2} x}=\tan^2 x$.

Here we add more examples without solutions for you to practice.

Example 3: Prove the identity: $\dfrac{\cos 2x}{\cos x- \sin x}=\cos x+\sin x$.

The following example is really common in exams.

Example 4: Solve $ \cos(2x)=\cos(x)$ when $0\leq x <2\pi$.

The following example is a bit challenging, but give it a try.

Example 5: Prove the identity: $\cos^4{x}=\dfrac{\cos(4x)}{8}+\dfrac{\cos(2x)}{2}+\dfrac{3}{8}$.

If you need help to understand the trig identities and Double Angle Formulas, contact us.

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