Answer
Verified
424.8k+ 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
Identify the feminine gender noun from the given sentence class 10 english CBSE
Your club organized a blood donation camp in your city class 10 english CBSE
Choose the correct meaning of the idiomphrase from class 10 english CBSE
Identify the neuter gender noun from the given sentence class 10 english CBSE
Choose the word which best expresses the meaning of class 10 english CBSE
Choose the word which is closest to the opposite in class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Discuss the main reasons for poverty in India
A Paragraph on Pollution in about 100-150 Words
Why is monsoon considered a unifying bond class 10 social science CBSE
What makes elections in India democratic class 11 social science CBSE
What does the term Genocidal War refer to class 12 social science CBSE
A weight hangs freely from the end of a spring A boy class 11 physics CBSE