The exact value of [tex]cos(\frac{11\pi }{8} )[/tex] is [tex]-\sqrt{\frac{2+\sqrt{2} }{4} }[/tex].
What is the exact value of trigonometric function?
The exact values of trigonometric functions are values of trigonometric functions of certain angles that can be expressed exactly using expressions containing real numbers and roots of real numbers. These values are also called analytic values of trigonometric functions or trigonometric values of special angles.
What is trigonometric function?
The trigonometric functions are also called the angle functions, which relates the angles and the ratios of the sides of a right angle triangle.
According to the given question.
We have a trigonometric function
[tex]cos(\frac{11\pi }{8} )[/tex]
We have to find the exact value of the above trigonometric function.
since,
[tex]\frac{11\pi }{8}[/tex] can be written as [tex](\pi +\frac{3}{8\pi } )[/tex]
[tex]\implies cos(\frac{11\pi }{8} ) = cos(\pi +\frac{3}{8}\pi )[/tex]
And we know that,
Cos(π + θ) = -cosθ
Because, π + θ lies in third quadrant and in third quadrat cos gives negative values.
Also,
[tex]cos\frac{\alpha }{2} = \pm\sqrt{\frac{1+cos\alpha }{2} }[/tex] ....(i)
Therefore,
[tex]cos(\pi +\frac{3}{8} \pi )[/tex]
[tex]= -cos(\frac{3}{8} \pi )[/tex]
[tex]= -cos(\frac{3}{2(4)} \pi )[/tex]
[tex]= -\sqrt{\frac{1+cos\frac{\pi }{4} }{2} }[/tex]
[tex]= -\sqrt{\frac{1+\frac{1}{\sqrt{2} } }{2} }[/tex]
[tex]= -\sqrt{\frac{1+\frac{\sqrt{2} }{2} }{2} }[/tex]
[tex]= -\sqrt{\frac{2+\sqrt{2} }{4} }[/tex]
Hence, the exact value of [tex]cos(\frac{11\pi }{8} )[/tex] is [tex]-\sqrt{\frac{2+\sqrt{2} }{4} }[/tex].
Thus, option A is correct.
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