Question

How do I prove $\cos \left( 2\pi -\theta \right)=\cos \theta ?$

Answer 1

Hint: In the given question we have to prove that $\cos \left( 2\pi -\theta \right)$ is equal to $\cos \theta $. Apply the trigonometric formula of $\cos \left( a-b \right)$ for evaluating the given trigonometric function. Put the required value of $\cos $ and $\sin $ directly to get a solution.

Complete step-by-step solution:
In this question, the given trigonometric function is.
$\cos \left( 2\pi -\theta \right)$
By applying trigonometric formula of $\cos \left( a-b \right)$ which is $\cos \left( a-b \right)=\cos a\cos b+\sin a\sin b$
Let, the value of $a=2\pi $ and the value of $b=\theta $
Now, put the above values in the trigonometric formula, we get,
$\Rightarrow$$\cos \left( 2\pi -\theta \right)=\cos \left( 2\pi \right)\cos \left( \theta \right)+\sin \left( \pi \right)\sin \left( \theta \right)$
We know that the value of $\cos 2\pi $ is equal to $1.$
$\Rightarrow$$\cos \left( 2\pi -\theta \right)=\left( 1 \right)\cos \left( \theta \right)+\sin \left( 2\pi \right).\sin \left( \theta \right)$
Also, we know that the value of $\sin \left( 2\pi \right)$ is equal to $0$
$\Rightarrow$$\cos \left( 2\pi -\theta \right)=\left( 1 \right)\cos \left( \theta \right)+\left( 0 \right)\sin \left( \theta \right)$
Now, multiply $1$ and $\cos \theta $ we get,
$\Rightarrow$$\cos \left( 2\pi -\theta \right)=\cos \theta +\left( 0 \right)\sin \left( \theta \right)$
Now, multiply $0$ and $\sin \theta $ we get
$\Rightarrow$$\cos \left( 2\pi -\theta \right)=\cos \theta +0$
Add $\cos \theta $ and $0$
$\Rightarrow$$\cos \left( 2\pi -\theta \right)=\cos \theta $

Hence, it is proven that $\cos \left( 2\pi -\theta \right)=\cos \theta $

Note: The trigonometric functions are real functions which relate an angle of a right angled triangle to ratios of two side lengths.
There are six basic trigonometric functions. They are sine, cosine, tangent, cotangent, secant and cosecant.
Some sine and cosine addition and subtraction formulas are as follows:
Addition formula for sine.
$\sin \left( a+b \right)=\sin a\cos b+\cos a\sin b$
Subtraction formula for sine
$\sin \left( a-b \right)=\sin a\cos b-\cos a\sin b$
Addition formula for cosine.
$\cos \left( a+b \right)=\cos a\cos b-\sin a\sin b$
Subtraction formula for cosine
$\cos \left( a-b \right)=\cos a\cos b+\sin a\sin b$