Inverse Trig: Choosing One Angle

Idea

Sine takes an angle and gives a number. The inverse should do the opposite: take a number and give the angle.

There is a problem, and you can find it yourself. Drag to $30$ degrees below and note the sin readout: $1/2$. Now look at the second marked point. It reads $1/2$ too - and past a full turn there are more.

"The angle whose sine is $\dfrac{1}{2}$" names many angles: $30^\circ$, $150^\circ$, $390^\circ$, and on without end. A function must give exactly one answer, so we have to make a choice. The rule below is that choice.

Rule

An inverse exists only for a function that is one-to-one. Trigonometric functions are not, so we restrict each to a stretch where it is, and invert only that stretch:

The restricted stretch becomes the range of the inverse. Every inverse trigonometric function is defined by choosing that window.

The same function is written two ways: $\arcsin x$ and $\sin^{-1} x$ mean exactly the same thing. In that second form the $-1$ is not an exponent - it marks the inverse function, not a reciprocal:

The reciprocal of sine has its own name, $\csc x$. Mixing the two is the most expensive notation error in this topic.

How it is used

Read the notation out loud: $\arcsin x$ means "the angle whose sine is $x$". A normal trig function goes from an angle to a number. An inverse trig function goes from a number to an angle.

So every value you already know can be read backwards.

There is a second use, in algebra. If $x$ is stuck inside a sine, arcsin is what gets it out. This is how you find the inverse of a function such as $f(x) = 3\sin x - 4$.

Now use the definition in both directions. $\arcsin\dfrac{1}{2}$ asks: which angle inside the window has sine $\dfrac{1}{2}$? The answer is $\dfrac{\pi}{6}$. The angle $150^\circ$ also has sine $\dfrac{1}{2}$, but it is outside the window, so it is not the answer.

We choose the ranges, but the domains are fixed for us. Sine only ever gives values between $-1$ and $1$. So $\arcsin 2$ has no meaning: no angle has sine $2$. Tangent gives every real number, so $\arctan$ accepts every real number.

This explains something surprising: $\arcsin(\sin x)$ does not always give back $x$.

Put in $x = 150^\circ$ and you get $30^\circ$. You get the angle inside the window, not the one you started with.

The other order is safe. $\sin(\arcsin x) = x$ is always true, because $x$ was already a valid sine value.

Explore

Drag along the green arc from $-90$ to $90$ degrees. Inside it, every sine value from $-1$ to $1$ appears exactly once - the badge under the circle says whether you are inside or outside the window.

The point starts at $150$ degrees, one of the many angles with sine $\dfrac{1}{2}$. It is outside the window, so of all those angles, arcsin keeps only $30$ degrees.