Question

How do you evaluate $\cos \left(-420\right)$?

Answer 1

Hint: Here we have to evaluate the given term as by using the formula. Also, we split the given cos term and we compare the formula. Finally we get the required answer.

Formula used: $\cos (2\pi +x)=\cos x$
$\cos (-x)=\cos x$

Complete step-by-step solution:
We need to evaluate the value of $\cos (-420)$.
Now we have to use the formula and we get, $\cos (-x)=\cos x$
Thus we can write it as, $\cos (-420)=\cos 420$
In order to solve it further, we use the formula:
$\cos (2\pi +x)=\cos x$ , where $$\pi =180$$ and $$x$$is any real number.
We further simplify $420$ to place it in the formula:
$420=2\left(180\right)+60$
This is equal to $420=2\pi +60$
Now, we can write it as,
$\cos (420)=\cos (2\pi +60)$
Using our formula$\cos (2\pi +x)=\cos x$, we get:
$\Rightarrow \cos (420)=\cos (60)$
Now, we know that$\cos 60={\displaystyle \frac{1}{2}}$
Therefore, $\cos 420=\cos 60={\displaystyle \frac{1}{2}}$

Thus, $\cos \left(-420\right)={\displaystyle \frac{1}{2}}$

Note: To the right and above of the origin, we have the positive side, while to the left and below the origin we have the negative side.

If we move anti-clockwise from the origin, then we find the first quadrant on the right upper side.
Moving anti –clockwise, the second quadrant is the one on the left, and then as we move below we find the third quadrant on the left below the number line followed by the fourth and last quadrant on the right below the number line.
In order to solve problems like these, we should be careful with the signs and pay attention to the quadrant that the trigonometric function falls in.
For example in the first quadrant, all trigonometric functions are positive, while in the second quadrant only sin and cosec is positive, while others are negative.
In the third quadrant tan and cot are positive while others are negative and in the final quadrant cos and sec are positive while all others are negative.