Question

Evaluate \[\cos \left( \pi -x \right)\]?

Answer 1

Hint: In this problem, we have to evaluate and find the answer for the given trigonometric expression. We can use trigonometric identity to evaluate the given trigonometric expression. We know that the trigonometric identity, \[\cos \left( A-B \right)=\cos A\cos B+\sin A\sin B\]. We can substitute the values of A and B by comparing the given expression and the trigonometric identity, to get an evaluated form.

Complete step by step answer:
We know that the given expression to be evaluated is,
\[\cos \left( \pi -x \right)\]
We can now use the trigonometric identity to evaluate the problem.
Here we can use the trigonometric identity,
\[\cos \left( A-B \right)=\cos A\cos B+\sin A\sin B\]
We can now find A and B to be substituted in the above trigonometric identity.
We can see that, \[A=\pi ,B=x\]
We can now substitute the above values in the trigonometric identity, we get
\[\Rightarrow \cos \left( \pi -x \right)=\cos \pi \cos x+\sin \pi \sin x\] …… (1)
We can now substitute the values for the required angles.
 We know that, \[\cos \pi =-1\] and \[\sin \pi =0\].
We can substitute the above values in (1), we get
\[\Rightarrow \cos \left( \pi -x \right)=-1\left( \cos x \right)+0\left( \sin x \right)\]
We can simplify the above step as the first term gets multiplied and the second term is 0, as it is multiplied to 0, we get
\[\Rightarrow \cos \left( \pi -x \right)=-\cos x\]

Therefore, the evaluated form is \[-\cos x\].

Note: Students make mistakes while writing the correct formula, we should concentrate while writing the sine and the cosine in the correct place and the addition or subtraction symbol should be correct according to the formula. We should also know some trigonometric degree values, like \[\cos \pi =-1\] and \[\sin \pi =0\], which are used in these types of problems.