Question

How do you find the exact value of$\cos 750^\circ $?

Answer 1

Hint: We already know the angles that have an exact expression for the trigonometry ratio. And those angles are $0^\circ ,30^\circ ,45^\circ ,60^\circ ,90^\circ $. There are lots more angles but not all angles have exact expressions involving nothing more than square-roots. So, we have to convert the angle between values 0 to 90. In this question, we will convert $750^\circ $between angle $0^\circ $ to $90^\circ $ and then substitute the value of trigonometry function at that angle.

Complete step-by-step answer:
Here, we want to find the value of$\cos 750^\circ $.
We have to convert the angle between values $0^\circ $ to $90^\circ $. As we know the value of those angles.
For that remove full rotation of $360^\circ $ until the angle between $0^\circ $ to $90^\circ $.
Now, let us convert $750^\circ $.
$ \Rightarrow 750^\circ = \left( {2 \times 360^\circ } \right) + 30^\circ $
So, we can write$\cos 750^\circ = \cos 30^\circ $.
Now, we already know the value of $\cos 30^\circ $.
 $ \Rightarrow \cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$
Therefore, the value of $\cos 750^\circ $ is:
$ \Rightarrow \cos 750^\circ = \dfrac{{\sqrt 3 }}{2}$
In decimal form, it is written as

$ \Rightarrow \cos 750^\circ = 0.86602$

Note:
To find the exact value of trigonometry functions without using a calculator has the following steps:
Assume that we want to find the exact value of f(x). Where f is any of the six trigonometric functions.
If the angle is negative, we first use a formula for negative angles such as $\sin \left( { - x} \right) = - \sin x,\cos \left( { - x} \right) = \cos x$, etc.
We locate the terminal side of the angle given in the question using a positive angle t.
The function value will be negative or positive is determined using the quadrant.
If the angle given in the question is in the first quadrant then the above step is not needed.
The value of $\sin 0^\circ = 0$
The value of $\sin 30^\circ = \dfrac{1}{2}$
The value of $\sin 45^\circ = \dfrac{1}{{\sqrt 2 }}$
The value of $\sin 60^\circ = \dfrac{{\sqrt 3 }}{2}$
The value of $\sin 90^\circ = 1$
The value of $\cos 0^\circ = 1$
The value of $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$
The value of $\cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$
The value of $\cos 60^\circ = \dfrac{1}{2}$
The value of $\cos 90^\circ = 0$
The value of $\tan 0^\circ = 0$
The value of $\tan 30^\circ = \dfrac{1}{{\sqrt 3 }}$
The value of $\tan 45^\circ = 1$
The value of $\tan 60^\circ = \sqrt 3 $
The value of $\tan 90^\circ $ is not defined.