New Zealand
Level 8 - NCEA Level 3

# Evaluating Inverse Trigonometric Functions

Lesson

Most calculators that have the trigonometric functions can also be used to evaluate the inverse trigonometric functions, directly.

For the $\sin$sin, $\cos$cos and $\tan$tan functions on most devices, one presses the '$2$2nd function' or 'shift' key and then the appropriate trigonometric function button to access the inverse function.

For example, if we need to find an angle whose sine is $0.25$0.25, we press 'shift', 'sin', $0.25$0.25, 'execute' (or the equivalent commands depending on the particular calculator being used) and the result, $14.47751219$14.47751219 in degrees should appear. For most practical purposes, we would round this number to at most four decimal places, bearing in mind that $1$1 second of arc is $\frac{1}{3600}^\circ$13600° or approximately $0.0003^\circ$0.0003°.

In the past, before electronic calculating machines were available, tables of trigonometric functions were used. A table of sines would show the sine of any angle between $0$0 and $90^\circ$90°, expressed to four decimal places. To find the inverse sine of, say $0.25$0.25, one would look for this number in the body of the table and, doing the lookup in reverse, deduce that it is the sine of $14.4775^\circ$14.4775°.

To find the inverses of values of the secant, cosecant and cotangent functions, which are usually not available on a calculator, we first write them as reciprocals of the cosine, sine and tangent functions and then proceed as before.

For example, if $\csc\theta=25.0018$cscθ=25.0018 and we require a value of $\theta$θ, we could write $\theta=\csc^{-1}(25.0018)$θ=csc1(25.0018) but this is not something most calculators can evaluate directly. Instead, we would go through the following reasoning:

 $\csc\theta$cscθ $=$= $25.0018$25.0018 $\frac{1}{\sin\theta}$1sinθ​ $=$= $25.0018$25.0018 $\sin\theta$sinθ $=$= $\frac{1}{25.0018}$125.0018​ $=$= $0.3999712021$0.3999712021 $\theta$θ $=$= $\sin^{-1}(0.3999712021)$sin−1(0.3999712021) $=$= $23.5764^\circ$23.5764°

We kept all the decimal digits in the calculation until the final step to avoid roundoff errors. If the number $0.3999712021$0.3999712021 had been rounded to four decimal places, it would become $0.4000$0.4000 and then $\sin^{-1}(0.4)=23.5782^\circ$sin1(0.4)=23.5782° which is different from the more accurate answer by about $6.5$6.5 seconds of arc.

Because the functions $\tan$tan, $\cot$cot, $\csc$csc and $\sec$sec have vertical asymptotes, a small change in the value of $\theta$θ, the argument of the function, when $\theta$θ is close to an asymptote, causes a relatively large change in the function value. Conversely, near an asymptote function values are large and a difference between them can result from very slightly different angles.

For this reason, calculations involving $\tan$tan$\cot$cot$\csc$csc and $\sec$sec and their inverses near asymptotes have to be treated with caution.

For example, $\tan87^\circ=19.08$tan87°=19.08, $\tan88^\circ=28.64$tan88°=28.64 and $\tan89^\circ=57.29$tan89°=57.29. We see that a small angle difference leads to increasing differences in the function values as the angle nears $90^\circ$90°. In the same way, if we were to evaluate $\arctan40$arctan40 and $\arctan50$arctan50 by calculator, we would obtain the results $88.568^\circ$88.568° and $88.852^\circ$88.852° respectively, a difference of only $0.286^\circ$0.286°.

A calculator returns single values of the inverse functions. But, when defined on their full domains, the trigonometric functions are periodic and thus give the same function value for many different values of the domain variable.

For example, given that $\cos\beta=-0.665$cosβ=0.665, find several solutions for the angle $\beta$β. By calculator, we find $\beta=\arccos(-0.665)=131.682^\circ$β=arccos(0.665)=131.682°. This is a 'second quadrant' angle, related to $180-131.682=48.318^\circ$180131.682=48.318° in the first quadrant. Since cosine is also negative in the third quadrant, there must be another angle, $180+48.318=228.318^\circ$180+48.318=228.318° with the same cosine.

Further solutions are obtained by adding multiples of $360^\circ$360° to these. The following representation should make this clear.

#### Example

Evaluate an inverse trigonometric function in order to find four solutions to the equation $\sec\theta=-4.35$secθ=4.35.

We rewrite this as $\frac{1}{\cos\theta}=-4.35$1cosθ=4.35. Then, $\cos\theta=\frac{1}{-4.35}$cosθ=14.35. Thus, $\theta=\arccos\frac{1}{-4.35}=103.29^\circ$θ=arccos14.35=103.29° is a solution. This angle is related to the first quadrant angle $13.29^\circ$13.29° and so, the third quadrant solution is $180+13.29=193.29^\circ$180+13.29=193.29°.

By subtracting $360^\circ$360° from each of these solutions we have
$\theta=103.29^\circ,193.29^\circ,-256.71^\circ,-166.71^\circ$θ=103.29°,193.29°,256.71°,166.71°.

#### Worked Examples

##### Question 1

Find the value of $\theta=\sin^{-1}\left(0.94489809\right)$θ=sin1(0.94489809) in decimal degrees.

##### Question 2

Find the value of $\theta=\csc^{-1}$θ=csc1$\left(-1.5275677\right)$(1.5275677) in decimal degrees.

##### Question 3

Find the real number value of $\theta=\sin^{-1}\left(0.46205386\right)$θ=sin1(0.46205386).