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

How do I prove cos⁡(2π−θ)=cos⁡θ?\cos \left( 2\pi -\theta \right)=\cos \theta ?

Explanation

Solution

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

Complete step-by-step solution:
In this question, the given trigonometric function is.
cos⁡(2π−θ)\cos \left( 2\pi -\theta \right)
By applying trigonometric formula of cos⁡(a−b)\cos \left( a-b \right) which is cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos \left( a-b \right)=\cos a\cos b+\sin a\sin b
Let, the value of a=2πa=2\pi and the value of b=θb=\theta
Now, put the above values in the trigonometric formula, we get,
⇒\Rightarrow cos⁡(2π−θ)=cos⁡(2π)cos⁡(θ)+sin⁡(π)sin⁡(θ)\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⁡2π\cos 2\pi is equal to 1.1.
⇒\Rightarrow cos⁡(2π−θ)=(1)cos⁡(θ)+sin⁡(2π).sin⁡(θ)\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⁡(2π)\sin \left( 2\pi \right) is equal to 00
⇒\Rightarrow cos⁡(2π−θ)=(1)cos⁡(θ)+(0)sin⁡(θ)\cos \left( 2\pi -\theta \right)=\left( 1 \right)\cos \left( \theta \right)+\left( 0 \right)\sin \left( \theta \right)
Now, multiply 11 and cos⁡θ\cos \theta we get,
⇒\Rightarrow cos⁡(2π−θ)=cos⁡θ+(0)sin⁡(θ)\cos \left( 2\pi -\theta \right)=\cos \theta +\left( 0 \right)\sin \left( \theta \right)
Now, multiply 00 and sin⁡θ\sin \theta we get
⇒\Rightarrow cos⁡(2π−θ)=cos⁡θ+0\cos \left( 2\pi -\theta \right)=\cos \theta +0
Add cos⁡θ\cos \theta and 00
⇒\Rightarrow cos⁡(2π−θ)=cos⁡θ\cos \left( 2\pi -\theta \right)=\cos \theta

Hence, it is proven that cos⁡(2π−θ)=cos⁡θ\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⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b\sin \left( a+b \right)=\sin a\cos b+\cos a\sin b
Subtraction formula for sine
sin⁡(a−b)=sin⁡acos⁡b−cos⁡asin⁡b\sin \left( a-b \right)=\sin a\cos b-\cos a\sin b
Addition formula for cosine.
cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos \left( a+b \right)=\cos a\cos b-\sin a\sin b
Subtraction formula for cosine
cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos \left( a-b \right)=\cos a\cos b+\sin a\sin b