Question

How do I find the value of $\cos ( - {240^ \circ })$?

Answer 1

Hint: Here we will use the negative $\theta $ property of cosine and then we will use the trigonometric ratios property of cosine to get the final answer.

Formula used: $\cos ( - \theta ) = \cos (\theta )$
$\cos ({180^ \circ } + \theta ) = - \cos \theta $

Complete step-by-step solution:
We have the given question as:
$ \Rightarrow \cos ( - {240^ \circ })$
Now we know that $\cos ( - \theta ) = \cos (\theta )$ therefore on using this formula on the given term we get:
$ \Rightarrow \cos ({240^ \circ })$
Now since there is no direct formula for getting the value of the angle, we will split it, since $240 = 180 + 60$ we will substitute it in the given term.
$ \Rightarrow \cos ({180^ \circ } + {60^ \circ })$
Now the above expression is in the form of $\cos ({180^ \circ } + \theta )$, since we know that the value of $\cos ({180^ \circ } + \theta )$ is $ - \cos \theta $, we can write the given expression as:
$ \Rightarrow - \cos ({60^ \circ })$
Now from the trigonometric table we know that the value of $\cos ({60^ \circ }) = \dfrac{1}{2}$, therefore we get:
$ \Rightarrow - \dfrac{1}{2}$

Therefore, we can conclude that $\cos ( - {240^ \circ }) = - \dfrac{1}{2}$.

Note: It is to be remembered which trigonometric functions are positive and negative in what quadrants.
The formula used over here is for $\cos ({180^ \circ } + \theta )$, the other formulas for the sine and cosine should be remembered.
When you add ${180^ \circ }$ to any angle, its position on the graph reverses, and whenever you add ${360^ \circ }$ to any angle, it reaches the same point after a complete rotation.
Basic trigonometric formulas should be remembered to solve these types of sums.
Since in this equation we had the angle as ${180^ \circ } + \theta $ we were able to use the formula directly, in other cases when there is addition of any two angles the addition-subtraction of angles property should be remembered and should be substituted to get the primitive sine, cosine and tan values.