Answer
397.5k+ 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.
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
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)