Question

What is the value of \[\cos (\dfrac{\pi }{8})\]?

Answer 1

Hint: In order to determine the value of \[\cos (\dfrac{\pi }{8})\] can be calculated by using the double identity formula,\[cos2X = 2{\cos ^2}X - 1\]. The degrees can be represented in terms of as follows:
\[{0^ \circ } = 0\],\[{30^ \circ } = \dfrac{\pi }{6}\],\[{45^ \circ } = \dfrac{\pi }{4}\],\[{60^ \circ } = \dfrac{\pi }{3}\],\[{90^ \circ } = \dfrac{\pi }{2}\] and \[{180^ \circ } = \pi \]
We can use the trigonometric ratio table to find out their value afterwards.

Complete step-by-step answer:
Trigonometry is a branch of mathematics that investigates the relationship between triangle side lengths and angles. In the given problem, we have to solve the cosine or cos function.
Let us assume \[X = \dfrac{\pi }{8}\]
Multiplying the above equation by two on both the sides, we get,
\[2X = \dfrac{\pi }{8}(2)\]
\[2X = \dfrac{\pi }{4}\]
Now we use the identity formula \[cos2X = 2{\cos ^2}X - 1\]and simplify as follows by substituting the value:
\[\cos (\dfrac{\pi }{4}) = 2{\cos ^2}X - 1\]
Taking one on the left side of equation, we get,
\[{\cos ^2}X = \cos (\dfrac{\pi }{4}) + 1\]
Now with help of trigonometric ratio table, we get,
\[\cos (\dfrac{\pi }{4}) = \cos ({45^ \circ }) = \dfrac{1}{{\sqrt 2 }}\]
Substituting the value, we get,
\[2{\cos ^2}X = \cos ({45^ \circ }) + 1\]
\[2{\cos ^2}X = \dfrac{1}{{\sqrt 2 }} + 1\]
Taking common denominator on right hand side, we get,
\[2{\cos ^2}X = \dfrac{{1 + \sqrt 2 }}{{\sqrt 2 }}\]
Now multiplying the numerator and denominator by \[\sqrt 2 \]on right-hand side, we get,
\[2{\cos ^2}X = \sqrt 2 (\dfrac{{1 + \sqrt 2 }}{{\sqrt 2 }})\]
\[2{\cos ^2}X = \dfrac{{1(\sqrt 2 ) + \sqrt 2 (\sqrt 2 )}}{{\sqrt 2 (\sqrt 2 )}}\]
\[2{\cos ^2}X = \dfrac{{\sqrt 2 + 2}}{2}\]
Taking two on the right-hand side by dividing, we get,
\[{\cos ^2}X = \dfrac{{\sqrt 2 + 2}}{{2(2)}}\]
\[{\cos ^2}X = \dfrac{{\sqrt 2 + 2}}{4}\]
Taking square root on both the sides of equation, we get,
\[\cos X = \sqrt {\dfrac{{\sqrt 2 + 2}}{4}} \]
\[\cos X = \dfrac{{\sqrt {2 + \sqrt 2 } }}{2}\]
Since, \[X = \dfrac{\pi }{8}\], we get,
Hence, the value of \[\cos (\dfrac{\pi }{8}) = \dfrac{{\sqrt {2 + \sqrt 2 } }}{2}\].

Note: Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. We have to use variables ‘x’ for the degree of the function to simplify the equation. Variables are generally alphabets used to denote unknown numerical value in algebraic equations. The cofunction of sine is cosine. The cofunction of cosine is sine. The cofunction of tangent is cotangent.