Cos3x Formula in terms of Cosx, Sinx [with Proof]

Cos3x formula in terms of cosx is as follows: 4cos³x-3cos x. In this post, we will also derive the cos3x formula in terms of sinx and give some examples. These formulas are very useful to solve trigonometric equations and simplify trigonometric expressions.

Cos3x Formula

Cos3x is the cosine function of a triple angle 3x. The formula of cos 3x in terms of cos x is given by the following identity:

cos3x =4cos³x-3cos x.

Cos3x in Terms of Cosx

To derive the cos3x formula, we will use the following trigonometric formulas:

1. $\cos(a+b)$ $=\cos a\cos b -\sin a \sin b$
2. $\sin 2x=2\sin x \cos x$
3. $\cos 2x=1-2\sin^2x$

Below are the steps to be followed in order to establish the formula of cos 3x.

Step 1: At the first step, we will write 3x as 2x+x, and then we will apply the above formula 1. By doing so, we get that

$\cos 3x=\cos(2x+x)$

$=\cos 2x \cos x -\sin 2x \sin x$

Step 2: Next, we will apply the formulas of sin2x and cos2x which are given in (2) and (3) above.  Thus, we obtain that

$\cos 3x=(1-2\sin^2x) \cos x$ $-(2\sin x\cos x)\sin x$

$=\cos x -2\sin^2 x \cos x-2\sin^2 x\cos x$

$=\cos x-4\sin^2x\cos x$

$=\cos x-4(1-\cos^2x)\cos x$ as we know that $\sin^2x=1-\cos^2x$.

$=\cos x -4\cos x+4\cos^3x$

$=4\cos^3 x -3\cos x$

Hence, the formula of $\cos 3x$ is $4\cos^3x-3\cos x$.

Question: Find the value cos135 degrees.

Answer:

In the above formula of cos3x, that is, $\cos 3x=4\cos^3x-3\cos x$, we put $x=45^\circ$. Thus, we obtain that

$\cos 135^\circ= 4\cos^3 45^\circ-3\cos 45^\circ$

$=4(\frac{1}{\sqrt{2}})^3-3\times \frac{1}{\sqrt{2}}$ as we know that $\cos 45^\circ=1/\sqrt{2}$

$=\frac{4}{2\sqrt{2}}-\frac{3}{\sqrt{2}}$

$=\frac{4-6}{2\sqrt{2}}$

$=\frac{-2}{2\sqrt{2}}$

$=-\frac{1}{\sqrt{2}}$

Thus, the value of cos135 degrees is equal to -1/root(2).

Cos3x in Terms of Sinx

Now, we will find the formula of cos3x in terms of sinx using the Pythagorean identity $\cos^2x+\sin^2x=1$.

Now, $\cos3x=4\cos^3x-3\cos x$

$=4(\sqrt{1-\sin^2x})^3-3\sqrt{1-\sin^2x}$

$=4(1-\sin^2x)^{3/2}-3\sqrt{1-\sin^2x}$.

Also Read: 

Tan2x Formula in terms of tanx, sinx, cosx

sin3x Formula in terms of sinx

Simplify 1-sin^2(x)

Values of sin15, cos15, tan15

Values of sin75, cos75, tan75

Simplify cos(sin^{-1}x)

Simplify sin(cos^{-1}x)

FAQs