Question

How do I find the value of $\cos ( - 240)?$

Answer 1

Hint: As we know that the above question is related to trigonometry as cosine or $\cos $is a trigonometric ratio. Here we will use some of the basic trigonometric identities to solve the problem. We know that $\cos ( - \theta ) = \cos \theta $, we use this identity and then we break the value inside the brackets into sum of two angles and then apply their formula i.e. $\cos (a + b) = \cos (a)\cos (b) - \sin (a)\sin (b)$.

Complete step-by-step solution:
Here we have $\cos ( - 240)$.
By applying the identity $\cos ( - \theta ) = \cos \theta $, we can say that $cos( - 240) = \cos 240$.
Now we have to find the value of $\cos 240$, it can be written as the sum of two angles i.e. $\cos (180 + 60)$
We know the formula $\cos (a + b) = \cos (a)\cos (b) - \sin (a)\sin (b)$.
Applying the formula we get: $\cos 180\cos 60 - \sin 60\sin 180$.
We know that $\cos 180 = - 1,\cos 60 = \dfrac{1}{2}$ and the second part does not matter as $\sin 180 = 0$.
So we get: $\cos 240 = - 1 \times \dfrac{1}{2} = - \dfrac{1}{2}$.

Hence the required value of $\cos ( - 240) = - \dfrac{1}{2}$.

Note: In this question we have applied the sum formula because the given expression can be expressed as the sum of two angles. We should always know the values. We should also know how to write expressions as the difference formula i.e. $\cos (A - B) = \cos A\cos B + \sin A\sin B$. We can write $\cos (360 - 240) = \cos 120$ and we know that $\cos 120$ gives a negative value as it lies in the second quadrant.