Answer

Verified

339.9k+ views

**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.

Recently Updated Pages

The branch of science which deals with nature and natural class 10 physics CBSE

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Distinguish between the reserved forests and protected class 10 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give simple chemical tests to distinguish between the class 12 chemistry CBSE

Difference Between Plant Cell and Animal Cell

Which of the following books is not written by Harshavardhana class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

In which states of India are mango showers common What class 9 social science CBSE

What Made Mr Keesing Allow Anne to Talk in Class class 10 english CBSE