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**Hint:**In this question, the value of \[\cos \left( {\dfrac{\pi }{4}} \right)\] is obtained first converting the radian angle to the degree angle and then, considering this as the triangle that has the respective angle and then find the value of the corresponding angle. The last step is to rationalize the fraction obtained.

The radian measure is defined as the ratio of the length of the circular arc to the radius of the arc, the measure of the angle is determined by the rotation from the initial side to the final side, and the angle is measured in degrees and in trigonometry the degree measure is \[\dfrac{1}{{{{360}^{{\text{th}}}}}}\] of the complete rotation.

**Complete Step by Step solution:**

We have given the trigonometric ratio as \[\cos \left( {\dfrac{\pi }{4}} \right)\].

As we know the formula to convert the radian measure to degree measure is,

\[ \Rightarrow Degree\;measure = radian\;measure\left( {\dfrac{{180^\circ }}{\pi }} \right)\]

Convert \[\dfrac{\pi }{4}\] is converted degree measure as,

\[

\Rightarrow \theta = \dfrac{\pi }{4}\left( {\dfrac{{180^\circ }}{\pi }} \right) \\

\Rightarrow \theta = 45^\circ \\

\]

Thus, the required value of the degree measure is \[45^\circ \].

The value of \[\cos \left( {\dfrac{\pi }{4}} \right)\] is calculated as \[\cos \left( {45^\circ } \right)\] by considering a triangle in which one of the angle is\[45^\circ \]. Since, the ratio of cosine angle is equal to the ratio of base to hypotenuse, the value of \[\cos \left( {45^\circ } \right)\] is \[\dfrac{1}{{\sqrt 2 }}\].

Rationalize the fraction \[\dfrac{1}{{\sqrt 2 }}\] by multiplying the denominator and the numerator by \[\sqrt 2 \] as,

\[ \Rightarrow \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }} = \dfrac{{\sqrt 2 }}{2}\]

**Thus, the value of \[\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{{\sqrt 2 }}{2}\].**

**Note:**

As we know that the trigonometry is the part of calculus and the basic ratio of trigonometric are sine and cosine which have their application in sound and light wave theories. The trigonometric have vast applications in naval engineering such as to determine the height of the wave and the tide in the ocean.

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