calculator issue: radians or degrees for inverse trig

Question

It's a simple question but I am a little confused. The value of $cos^{-1} (-0.5)$ , is it 2.0943 or 120 ?

Answer 1

If you think of $\cos^{-1}(x)$ as an angle, then you can express the answer in either degrees or radians.

If you think of this as a real number, though, then it always takes on a value between 0 and $\pi$ (which is numerically equal to the angle in radians).

Answer 2

It helps to understand that there are several different functions called cosine. I find it useful to refer to "cos" (the thing for which $\cos^{-1}(0) = \pi/2$) and "cosd" for which cosd(90) = 0.

Your calculator (if you're lucky) will mean "cos" when you press the "cos" button; if you've got the option of "degrees/radians", then in "degrees" mode, the "cos" button is actually computing the function "cosd".

Answer 3

The principal value of $\cos^{-1}(-0.5)$ is $120^{\circ}$ or $\frac{2 \pi}{3} \approx 2.0943$ radians. Take your pick!

A couple of key points here.

First, it helps to have a feeling for what range of values to expect when you do an operation. If I'm taking $\cos^{-1}x$ I expect to get an angle. If the angle is measured in degrees, reasonable values are $0$ to $180$. If the angle is measure in radians, reasonable values are $0$ to $\pi \approx 3.14159265$.

For angles in general, $-180$ to $360$ would be resonable for degree measures, and $-3.14159265$ to $6.2831853$ or $-\pi$ to $2\pi$ would be reasonable for radian measures.

Second, though, any intuition should be verified by knowing what you're looking at. You should know whether you need to input angles in degrees or radians, and you should know whether the angles you get back are expressed in degrees or in radians.

Because if you don't know, your intuition may not be enough. What if you're calculating $\cos(\pi / 100)$? Here, $\pi/100$ is an angle in radians. You enter $0.031415926535$ and push the $\cos$ key. If you're in "radians" mode, the answer you get back is $\approx 0.999506560$, but if you're instead in "degrees" mode, the answer you get back is $\approx 0.999999849$, which is substantially a different answer, but not one that you could easily tell was wrong just by looking at it. "The angle is close to zero, $\cos$ should be close to one" wouldn't have been enough intuition to catch the error.

Answer 4

The answer depends on whether you are working in degrees or radians. It's just a case of units.

When we think about length, we have $1 \, \text{km} = 1,000\,\text{m}$.

When we talk about velocity we have $1 \, \text{km/h} = \tfrac{8}{15}\text{m/s}$.

When we talk about angles we have $180^{\circ} = \pi^c$.

However, when we are doing calculus: we must always use radians.

The usual definition of cosine is in terms of the ratio of the adjacent and the hypotenuse of a right-angled triangle. In this case, degrees or radians are fine. There is another "definition" of the cosine function which comes from the Taylor Series:

$$\cos x \sim 1 - \tfrac{1}{3!}x^3 + \tfrac{1}{5!}x^5 - \tfrac{1}{7!}x^7 + \cdots $$

This definition depends on differentiation, i.e. on using calculus. Hence: you must use radians.