Question

How do you find the exact value of $\cos \dfrac{{5\pi}}{4}$?

Answer 1

Hint: The angle used here is an obtuse angle, so we will first convert the given obtuse angle into an acute angle. To convert the obtuse angle into the acute angle, we will use the periodic trigonometric identities. Then we will substitute the value of the cosine function to get the required answer.

Formula used:
$\cos \left( {\pi + \theta } \right) = - \cos \theta $

Complete step by step solution:
 Here we need to find the value of the cosine of the given angle i.e. $\cos \left( {\dfrac{{5\pi }}{4}} \right)$
We will first convert the given obtuse angle into an acute angle.
We can write the angle inside the bracket as the sum of two angles.
$\cos \left( {\dfrac{{5\pi }}{4}} \right) = \cos \left( {\pi + \dfrac{\pi }{4}} \right)$
We know from the periodic trigonometric identities that $\cos \left( {\pi + \theta } \right) = - \cos \theta $.
Using the same trigonometric identities, we get
$ \Rightarrow \cos \left( {\dfrac{{5\pi }}{4}} \right) = - \cos \dfrac{\pi }{4}$
We know that the value of $\cos \dfrac{\pi }{4}$ is equal to $\dfrac{1}{{\sqrt 2 }}$. So we will substitute this value of cosine function here.
 $ \Rightarrow \cos \left( {\dfrac{{5\pi }}{4}} \right) = - \dfrac{1}{{\sqrt 2 }}$

Therefore, the exact value of $\cos \dfrac{{5pi}}{4}$ is equal to $ - \dfrac{1}{{\sqrt 2 }}$.

Additional information:
Different trigonometric functions have a different sign in every quadrant. In the first quadrant, all the functions are positive. In the second quadrant, only sine and cosecant functions are positive and other functions are negative. Similarly, in the third quadrant tangent and cotangent functions are positive, and in the fourth quadrant cosine and secant functions are positive and other functions are negative.

Note: Trigonometric identities are defined as the equalities which include the trigonometric functions and they are always true for every value of the occurring variables where both sides of the equality are well defined. These identities are used in the equation where different trigonometric functions are present. The given angle is greater than 180 degrees and less than 270 degrees, so it lies in the third quadrant. In the third quadrant cosine and secant are negative so we got the negative value of the given angle.