Question

How do you evaluate \[\cos \left( {\dfrac{{15\pi }}{4}} \right)\]?

Answer 1

Hint: We will use concepts of trigonometry to solve this problem. We will define the cosine function in detail while solving this problem and we will take a look at some standard results of trigonometry too. We will also define the terminology of a right-angled triangle.

Complete step by step solution:
In a right-angled triangle, the side with maximum length or the side opposite the right angle is called the hypotenuse.
In mathematics, trigonometry deals with ratios of sides of a right-angled triangle. So, in a right-angled triangle, the cosine of an angle is defined as, ratio of sides adjacent to the angle to length of the hypotenuse.
But the cosine of a right angle, i.e., cosine of \[{90^ \circ }\] is not defined.
Cosine value is positive in the range \[\left[ {\dfrac{{ - \pi }}{2},\dfrac{\pi }{2}} \right]\]. For any other angle which is not from this range, the cosine value is negative.
So, we can say that, cosine is positive in the first and fourth quadrants and negative in the second and third quadrants.
Now, the question given is \[\cos \left( {\dfrac{{15\pi }}{4}} \right)\]
We can write that as,
\[ \Rightarrow \cos \left( {\dfrac{{15\pi }}{4}} \right) = \cos \left( {\dfrac{{16\pi - \pi }}{4}} \right) = \cos \left( {4\pi - \dfrac{\pi }{4}} \right)\]
So, the angle \[4\pi - \dfrac{\pi }{4}\] belongs to the fourth quadrant. So, the cosine of this angle is positive.
And also, there is a formula which is \[\cos \left( {2n\pi \pm \theta } \right) = \cos \theta \]
So, according to this,
\[ \Rightarrow \cos \left( {4\pi - \dfrac{\pi }{4}} \right) = \cos \left( {2(2)\pi - \dfrac{\pi }{4}} \right) = \cos \dfrac{\pi }{4}\]
And we all know the standard result \[\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\]
We can conclude that, \[\cos \left( {\dfrac{{15\pi }}{4}} \right) = \dfrac{1}{{\sqrt 2 }}\].

Note:
We can further write it as, \[\dfrac{1}{{\sqrt 2 }} = \dfrac{{\sqrt 2 }}{2} = 0.707\] by rationalizing the denominator.
Also remember the identity \[\cos \left( {90 + \theta } \right) = - \sin \theta \] which are some standard results. We can also find the cosine of an angle by using the sine value of that angle too. Suppose that you know the value of \[\sin x\], then you can find a cosine of this angle by applying \[\cos x = \pm \sqrt {1 - {{\sin }^2}x} \].
This came from the identity \[{\cos ^2}x + {\sin ^2}x = 1\].