Definition of Inverse Trigonometric Functions
All the mathematical functions, right from the simplest to the most complex, have an inverse. In the world of mathematics, inverse primarily means the opposite. When it comes to addition, the inverse is subtraction and for multiplication, it is division. And talking about trigonometric functions, it is always inverse trigonometric functions.
Trigonometric functions can be defined as the functions of an angle and it is basically used for describing the relationship between two sets of numbers or variables. In modern mathematics, there are a total of six basic trigonometric functions: sine, cosine, tangent, secant, cosecant and cotangent. And we can express the inverse of these functions as sine, inverse cosine, inverse tangent, inverse secant, inverse cosecant and inverse cotangent.
Note that the inverse trigonometric relations are actually functions as for any given input, there is more than one output. Which means there is more than one angle whose sine, cosine, etc. is that number (for a given number). But the ranges of the inverse relations can be limited such that there is a one-to-one correspondence between the inputs and outputs of the inverse relations. Keeping these restricted ranges in mind, the inverse trigonometric relations become the inverse trigonometric functions.
Also, all the symbols for the inverse functions are different from the symbols for the inverse relations and the names of the functions are capitalised. These are the inverse functions: Arcsine, Arccosine, Arctangent, Arccosecant, Arcsecant, and Arccotangent. They are also expressed like: y = sin-1(x), y = cos-1(x), etc. The chart given below mentions the restricted ranges that are responsible for transforming the inverse relations into the inverse functions.
The inverse trigonometric functions are as good as the inverse trigonometric relations. But when it comes to inverse functions, it only gives one output per input–whichever angle lies within its range due to its restricted range. Therefore, it makes a one-to-one correspondence, making the inverse functions more practical.
The trigonometric functions can be called as ratios of the sides of a right triangle. We know that all right triangles adhere to the Pythagorean Theorem. So, the sides will be proportional as long as the angles of the two right triangles are the same. Due to this, the ratios of one side to another will always be the same. Check out this example.
As you can see, the angles of these triangles have the same measures, so their sides are proportional. Any ratio of one side to another will be the same for both triangles.
6/10 = 3/5
This way, by discovering that these ratios are the same for a right triangle with any size (as long as they have the same angle measure), the trigonometric functions were figured out. These functions relate one angle of a triangle to the ratio of two of its sides.
Now, since these ratios, when an angle (other than the right angle) of a right triangle and at least one side are known, we can easily find out the length of the other sides with the help of these ratios. And inversely, when we know the lengths of two sides, the angle measure can be found out.
These ratios can look difficult, but you can take the help of a mnemonic for keeping them straight. SOH CAH TOA is a helpful trick for remembering which ratio can be clubbed with which function.
Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent
The inverse trigonometric functions can be helpful for finding out the angle measure when at least two sides of a right triangle are known. For deciding which particular function needs to be used, we need to know what two sides are known. For instance, if we are aware of the hypotenuse and the side opposite the angle, we can use the inverse sine function. Or in a situation when we know the side opposite and the side adjacent to the angle, we can use the inverse tangent function.
We can use two methods for finding out an inverse trigonometric function. The first method is to use a table that has all the results for every ratio. But this process can be cumbersome. The other technique includes using a scientific calculator. In this method, the inverse functions for the sine, cosine and tangent can be calculated quickly.
Example 1: Find the measure of the angles below.
Solution: For part a, we can use the sine function and for part b, we can use the tangent function. Since both problems need us to solve for an angle, the inverse of each can be used.
a. sinx=7/25→sin−17/25=x→x= 16.26∘
b. tanx=40/9→tan−140/9=x→x= 77.32∘
Note that we know the trigonometric value tanθ=40/9 of the angle, but we have no clue about the angle. So, in this example, the inverse of the trigonometric function needs to be used for finding the measure of the angle.
The inverse of the tangent function is expressed as “tangent inverse”, but is also called the arctangent relation. The inverse of the cosine function is read as “cosine inverse”, but is also called the arccosine relation. Similarly, the inverse of the sine function is primarily read “sine inverse” and called the arcsine relation.
Example 2: In standard position, find the angle θ.
Solution: The tanθ=y/x or, in this example, tanθ=8/−11. By making use of this inverse tangent, we get tan−1−8/11=−36.03∘.
This signifies that the reference angle is 36.03∘. Also, this value we get of 36.03∘ is the angle you can observe if we move counterclockwise from the -x axis. For finding out the corresponding angle in the second quadrant (which is the same as though you started at the +x axis and moved counterclockwise), we have to subtract 36.03∘ from 180∘, yielding 143.97∘.
Keep in mind that inverse trigonometric functions are also useful for finding out the angle of depression or elevation.
Example 3: A new outdoor skating rink has just got set up near a local community center. Then, a light is fixed on a pole that is around 25 feet above the ground. The light needs to be positioned at an angle so that it illuminates the end of the skating rink. So, at what angle of depression should the light be installed if the end of the rink is 60 feet away from the pole?
Solution: In this case, we are not aware of the angle of depression, which is located outside of the triangle. We know that the angle of depression is equal to the angle of elevation. The pole where the light’s located is the opposite and is 25 feet high when it comes to the angle of elevation. Also, note that the length of the rink is the adjacent side and is 60 feet in length. So, for calculating the measure of the angle of elevation, we can use the trigonometric ratio for the tangent.
tanθ = 25/60
tanθ = 0.4166
tan−1 (tanθ) = tan−1 (0.4166)
Therefore, the angle of depression at which the light must be positioned for lighting up the rink is 22.6∘