Question

How do you find the exact value of $\cos \left( \dfrac{33\pi }{4} \right)$ ?

Answer 1

This given problem is a great example of trigonometric equations and functions. To solve this problem we need to have an in-depth knowledge of trigonometric functions. In problems like these we first need to convert the given angle in such a way that it reduces to the smallest possible formed angle. After that we need to find in which quadrant the angle formed is located. In our problem, the given trigonometric function is cosine, so if the angel falls in the first or fourth quadrant, the value will be positive. For all other quadrants the value will be negative.

Complete step by step solution:
Now we start off with the solution to our given problem by writing it as,
We first try to reduce the angle to the smallest possible value by factorisation. We can write the given equation as,

& \cos \left( \dfrac{33\pi }{4} \right) \\\ & =\cos \left( 8\pi +\dfrac{\pi }{4} \right) \\\ \end{aligned}$$ Now we know that, for locating an angle in the proper quadrant, we need to go around the coordinate axes. For one complete rotation the angle that we cover is $$2\pi $$ . Now for a given angle of $$8\pi $$ , we make $$4$$ complete rotations and after that we rotate an angle of $$\dfrac{\pi }{4}$$ . From this we can clearly understand that our resultant final angle lies in the first quadrant. Thus, for the given problem, our answer will be as simple as writing it as, $$\begin{aligned} & \cos \left( \dfrac{\pi }{4} \right) \\\ & =\dfrac{1}{\sqrt{2}} \\\ \end{aligned}$$ Thus, our answer to the problem is $$\dfrac{1}{\sqrt{2}}$$. **Note:** In problems like these we first of all need to reduce the given angle to its lowest possible form so that evaluation of the value becomes easier. After that we need to find out the right quadrant by checking out the rotation of the given angle. We must carefully handle the positive and negative values which may be a part of the given problem. Finally, we must remember all the possible values of the trigonometric functions for different angles.