Question

What is the value of $\cos \left( { - \theta } \right)$ in terms of $\cos \theta $ ?

Answer 1

Hint: The given question deals with simplification of a trigonometric composite function in terms of a simple trigonometric ratio. We will also make use of trigonometric formulae such as $\cos \left( {2\pi - x} \right) = \cos x$ to solve the problem.Basic algebraic rules and trigonometric identities are to be kept in mind while simplifying the given problem and proving the result given to us. We should also know about the periodicity of the trigonometric functions to tackle such types of problems.

Complete step by step answer:
In the given problem, we have to convert $\cos \left( { - \theta } \right)$ in terms of $\cos \theta $. This can be further used in many questions and problems as a direct result and has wide ranging applications. For getting to the desired result, we need to have a good grip over the basic trigonometric formulae and identities.

So, we have, $\cos \left( { - \theta } \right)$. We know that the trigonometric function cosine is a periodic function that repeats its value after an interval of $2\pi $ radians. So, adding or subtracting $2\pi $ radians from the angle of the trigonometric function does not make any difference to the value of the trigonometric function. So, we get,
$\cos \left( { - \theta } \right) = \cos \left( {2\pi - \theta } \right)$
Now, we know the trigonometric formula for cosine of angle in the fourth quadrant as $\cos \left( {2\pi - x} \right) = \cos x$. So, we have,
$\cos \left( { - \theta } \right) = \cos \left( {2\pi - \theta } \right)$
$ \therefore \cos \left( { - \theta } \right) = \cos \theta $

Hence, $\cos \left( { - \theta } \right)$ can be represented in terms of cosine as $\cos \theta $.

Note: Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart. Besides these simple trigonometric formulae, we should also have knowledge about the periodicity of trigonometric functions.The problem can also be solved keeping in mind that the cosine function is an even function. So, it gives the same output for positive and negative input of the same magnitude. Hence, $\cos \left( { - \theta } \right) = \cos \theta $.