Inverse Trigonometric Functions

Objectives

By the end of this lesson you will be able to

Knowing the value of sin(x), cos(x) or tan(x), use the unitary circle in order to look for possible values of x.
Explain why is it necessary to restrain the domain in order for sin(x), cos(x) and tan(x) to exist.
Define and graph trigonometric inverse functions.

Introduction

In the Unitary circle section and Other Trigonometric, we defined sine, cosine y tangent in the following manner:

In the Inverses Function we determine when two functions are inverses if for any value a,

In consequence, we may infer that the inverses of the trigonometric functions are defined in the following manner.

Use the following app of the circle in order to observe the angles and locations and the values of the respectives functions of sine, cosine, and its inverse
Utilice el cursor para mover el punto de color verde.

Este es un Applet de Java creado con GeoGebra desde www.geogebra.org – Java no parece estar instalado Java en el equipo. Se aconseja dirigirse a www.java.com

If sin (x) = 0.5, What's the value of x?
Solution:
sin (x) = 0.5 implies that x is the angle that correspond to the location with coordinate y = 0.5 on the unit circle.
Observing the image we found that the angles that satisfy this condition are 30° and 150°.
This condition is also satisfied for all angles that are coterminal with 30° and 150°. For example, 390°, 510°, ...

If cos (x) = -0.5, What's the value of x?
Solution:
cos (x) = -0.5 implies that x is the angle that correspond to the location with coordinate y = -0.5 on the unit circle.
Observing the image we found that the angles that satisfy this condition are 120° and 240°.
This condition is also satisfied for all angles that are coterminal with 120° and 240°. For example, 480°, 600°, ...

If sin (x) = 0, What's the value of x?
Solution:
sin (x) = 0 implies that x is the angle that correspond to the location with coordinate y = 0 on the unit circle.
Observing the image we found that the angles that satisfy this condition are 0° and 180°.
This condition is also satisfied for all angles that are coterminal with 0° and 180°. For example, 360°, 540°, ...

If tan (x) = 1, What's the value of x?
Solution:
tan (x) = 1 implies that x is the angle that correspond to the location with coordinate x = y on the unit circle.
Observing the image we found that the angles that satisfy this condition are 45° and 225°.
This condition is also satisfied for all angles that are coterminal with 45° and 225°. For example, 405°, 585°, ...


Example

Restraining the domain of the Trigonometric Functions in order to define their inverses

Just as we have observed in the previous examples of the previous sections, given a value of the trigonometric function it's possibke to find the angle that complies with that value

If we follow the process of the previous section we obtain the following tables and graphs for certain values

Restraining the domain of function sin(x)

sin(x)	0.00	0.50	0.71	0.87	1.00	0.87	0.71	0.50	0.00	-0.50	-0.71	-0.87	-1.00	-0.87	-0.71	-0.50	0.00	0.50	1.00
x	0	30	45	60	90	120	135	150	180	210	225	240	270	300	315	330	360	390	450

We analyze if the table complies with function:

In the table above we can see for the input 0, has more than one output therefore is not a function.

Graphically, in the image to the left, we can see an example of the points(0.71,45) y (0.71,135) belong to the graph in other words is not a function.

In order to understand this, we need to remember Inverse Functions we've seen that for a function to have an inverse it needs to be 1:1. The function sin(x) is not 1:1. Inorder to obtain the inverse of sin(x) we need to restrain its domain and interval where it''s 1:1.

The function sin(x) is 1:1 in the interval [-90°,90°] or in radians, [-^π/₂,^π/₂].

Restraining the domain of the function cos(x)

cos(x)	1.00	0.87	0.71	0.50	0.00	-0.50	-0.71	-0.87	-1.00	-0.87	-0.71	-0.50	0.00	0.50	0.71	0.87	1.00	0.87	0.00
x	0	30	45	60	90	120	135	150	180	210	225	240	270	300	315	330	360	390	450

Analyze if the graph and table correspond to the function

In the table above we can see for the input 1 has more than one output therefore is not a function.

Graphically, on the image on the left, we see for example the points (0.71,45) y (0.71,315) belong to the graph, meaning is not a function.

The explanation is that the function cos(x) is not 1:1. In order to obtain the inverse of sin(x) we need to restrain its domain where it is1:1.

The function cos(x) is 1:1 in the interval [0°,180°] or in radians, [0,π].

Restrain the domain of the function tan(x)

tan(x)	0.00	0.58	1.00	1.73	-1.73	-1.00	-0.58	0.00	0.58	1.00	1.73	-1.73	-1.00	-0.58	0.00	0.58
x	0	30	45	60	120	135	150	180	210	225	240	300	315	330	360	390

Analyze of the table and graph corresponding to the function:

In the table of above we can see the input 1, has more than one output, in other words is not a function.

We can appreciate the graph on the left side, let's take the following points as examples(0.71,45) y (0.71,315) which belong to the graph, meaning is not a function.

This is because cos(x) is not 1:1. In order to obtain the inverse of tan(x) we must restrain its domain in an interval that is 1:1.

The function tan(x) iss 1:1 in the interval [-90°,90°] or in radians (-^π/₂,^π/₂).

Definition

Trigonometric functions define themselves in the following manner

Tables and graphs of the trigonometric inverse funcions

Table

Graph

A table of values for the function sen(x) with domain [-^π/₂,^π/₂]:

x	$\frac{- π}{2}$	$\frac{- π}{3}$	$\frac{- π}{4}$	$\frac{- π}{6}$	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$
sin(x)	-1.00	-0.87	-0.71	-0.50	0.00	0.50	0.71	0.87	1.00

Hence, a table of values for the function sin^-1(x) is

x	-1.00	-0.87	-0.71	-0.50	0.00	0.50	0.71	0.87	1.00
sin^-1(x)	$\frac{3 π}{2}$	$\frac{5 π}{3}$	$\frac{7 π}{4}$	$\frac{11 π}{6}$	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$

Table

Graph

A table of values for the function cos(x) with domain [0,π]:

x	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{3 π}{4}$	$\frac{5 π}{6}$	π
cos(x)	1.00	0.87	0.71	0.50	0.00	-0.50	-0.71	-0.87	-1.00

Hence, a table of values for the function cos^-1(x) is

x	1.00	0.87	0.71	0.50	0.00	-0.50	-0.71	-0.87	-1.00
cos^-1(x)	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{3 π}{4}$	$\frac{5 π}{6}$	π

Tabla

A table of values for the function tan(x) with domain (-^π/₂,^π/₂):

x	$- \frac{π}{2}$	-1.54	-1.44	$- \frac{π}{3}$	$- \frac{π}{4}$	$- \frac{π}{6}$	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	1.44	1.54	$\frac{π}{2}$
tan(x)	Indefinite	-34.23	-7.697	-1.73	-1.00	-0.58	0.00	0.58	1.00	1.73	7.697	34.23	Indefinite

Hence, a table of values for the function tan^-1(x) is

x	-34.23	-7.697	-1.73	-1.00	-0.58	0.00	0.58	1.00	1.73	7.697	34.23
tan^-1(x)	-1.54	-1.44	$- \frac{π}{3}$	$- \frac{π}{4}$	$- \frac{π}{6}$	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	1.44	1.54

Graph

arctan

Inverse Trigonometric Functions
Haz Click en Cualquiera de las Funciones

In order to practice the trinometric inverse functions, click any of the buttons

Summary

Now that you have finished the section you are able to

Knowing the value of sin(x), cos(x) or tan(x), use the unitary circle in order to look for possible values of x.
Explain why is it necessary to restrain the domain in order for sin(x), cos(x) and tan(x) to exist.