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I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc.

What is the best way to go about this? So far, I have been trying to solve the problems intuitively (e.g. I ask myself what value of sine will give me $-\frac{\sqrt{3}}{{2}}$?), maybe drawing a triangle to help. Is there a better way to solve these problems?

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3 Answers 3

You need to know the basic values of the trigonometric functions, thus for example:

$$\sin\left(-\frac{\pi}3\right)=\sin\left(-\frac{2\pi}3\right)=\frac{\sqrt3}2\implies\arcsin\left(-\frac{\sqrt3}2\right)\in\left\{\;-\frac{\pi}3\;,\;-\frac{2\pi}3\;\right\}$$

so you must know where your values' range is.

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A combination of unit circle and "famous triangles" is perhaps the best strategy. Have drawing of the three famous triangles, $30,60,90$, with sides $1,2,\sqrt 3$, and $45,45,90$, with sides $1,1,\sqrt 2$ and $0,90,90$ with sides $0,1,1$. These will give you the reference angle you are looking for. Then for finding $z$ in $\arctan(a)=z,\arccos(a)=z,\arcsin(a)=z$ locate the axis on the unit circle and apply the definition to find the right quadrant for $z$. For sine the axis is the vertical $y$ axis and final angle is in fourth or first quadrant. For cosine the axis the horizontal $x$ axis and the angle is on first or second quadrant. For tangent the axis is $x=1$ line and the answer is on first or fourth quadrant.

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In other words, you know the values of $\arcsin x$/$\arctan x$/$\arccos x$ for some specific values of $x$. That's just fine. Inverse trigonometric functions are transcendental functions and with exceptions of a few well-known values, the result is not nicely expressible with elementary functions (you can use a calculator or any number of "approximations by hand" to get a numerical value for the angle, but that's not the same).

In addition to the classical angles of multiples of $30^\circ$, which have known values of trigonometric functions, you can use the half-angle formulas and addition theorems to get other angles (inverse functions can be computed by recognizing the half-angle and angle addition expressions and reducing the calculation to a simpler expression).

In fact, the angles you can construct by adding and halving of elementary angles ($45^\circ$ and $60^\circ$), are precisely the angles you can construct with a compass and a straightedge (constructible angles). Such constructions also mean that all the angles, for which you can get the trigonometric function without a calculator (or the inverse problem), have a nice geometric representation of how to carry out such a calculation. Construction with similar triangles and circles is thus a good idea if you have a more complicated expression that you think you can get analytically.

Otherwise, there's nothing wrong about knowing a few special values. We do that all the time.

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