Lecture 25 min.
Inverse trigonometric functions (circular functions, arc functions) are mathematical functions that are the inverses of the trigonometric functions. The inverse trigonometric functions usually include six functions:
the angle whose sine is equal to 
)
the angle whose cosine is equal to , and so on)
; in foreign literature 
)
; in foreign literature 
or 
)
)
; in foreign literature 
)The name of an inverse trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix "arc-" (from the Latin arcus, meaning arc). This is because, geometrically, the value of an inverse trigonometric function can be associated with the length of an arc of the unit circle (or the angle subtending that arc) corresponding to a particular segment. Thus, the ordinary sine lets us find the chord subtending a given arc of a circle, while the inverse function solves the opposite problem. The practice of denoting inverse trigonometric functions in this way was introduced by the Austrian mathematician Karl Scherffer (1716—1783) and became established thanks to Lagrange. Daniel Bernoulli was the first to use a special symbol for an inverse trigonometric function, in 1729. Until the end of the 19th century, the English and German schools of mathematics proposed other notations: 
but they did not catch on. Only occasionally, in foreign literature as well as on scientific and engineering calculators, are notations such as sin−1, cos−1 used for arcsine, arccosine, and so on; this notation is considered not very convenient, since it may be confused with raising a function to the power −1.
Trigonometric functions are periodic, so the functions inverse to them are multivalued. That is, the value of an arc function is a set of angles (arcs) for which the corresponding direct trigonometric function equals a given number. For example, 
means the set of angles 
whose sine is equal to 
. From the set of values of each arc function, its principal values are singled out (see the graphs of the principal values of the arc functions below), and these are what is usually meant when speaking of the arcsine, arccosine, and so on.
In the general case, provided that 
, all solutions of the equation 
can be represented as 
Trigonometric functions first arose in connection with research in astronomy and geometry. Ratios of segments in a triangle and a circle, which are essentially trigonometric functions, already appear in the 3rd century BC in the works of the mathematicians of Ancient Greece — Euclid, Archimedes, Apollonius of Perga, and others. Euclid Archimedes In the following period, mathematics was for a long time developed most actively by Indian and Arab scholars. In Aryabhata's works on astronomy the term "ardhajiva" appears. Later the shorter name "jiva" became established, and when mathematical terms were translated in the 12th century, this word was replaced by the Latin "sinus". Of fundamental importance was Ptolemy's compilation of the first table of sines (for a long time it was called a table of chords): a practical tool appeared for solving a number of applied problems, above all problems of astronomy. The word cosine is an abbreviation of the Latin expression "complementi sinus" (the sine of the complement). Tangents arose in connection with solving the problem of determining the length of a shadow. The tangent (as well as the cotangent, secant, and cosecant) was introduced in the 10th century by Abu al-Wafa, who also compiled the first tables for finding tangents and cotangents. However, these discoveries remained unknown to European scholars for a long time, and tangents were rediscovered in the 14th century by T. Bradwardine, and later by the astronomer Regiomontanus. The first author to use special symbols for the inverse trigonometric functions was Bernoulli. In 1729 and 1736 he wrote as and at instead of arcsin and arctan, respectively. The modern notations arcsin and arctan appeared in 1772 in the works of the Viennese mathematician Scherffer and the famous French scientist Lagrange. The prefix "arc" comes from the Latin "arcus" (bow, arc), which agrees fully with the meaning of the concept: arcsin x, for example, is the angle (or one can say the arc) whose sine is equal to x
First, remember the basic pair
There are four functions:
y=arcsin x
y=arccos x
y=arctan x
y=arccot x
It is convenient to divide them into two pairs:
The sine pair: (arcsin, arccos)
→ the graphs are restricted in x: only ([-1;1]).
The tangent pair: (arctan, arccot)
→ the graphs stretch across the whole x range, but are restricted in y.
arcsinx: "a smile"
arccosx: "an upside-down arcsin"
arctanx: "rises and hits the ceiling"
arccotx: "the arctangent in reverse"
To a first approximation, a glide path resembles an inverse trigonometric function such as the arcsine or arctangent (a glide path is the flight trajectory of an aircraft (airplane, helicopter, glider, parachutist) along which it descends, including immediately before landing. ). Of course, describing the flight of a UAV requires a more complex spatial mathematical model
Here vx, vy, vz are the projections of the velocity vector of the vehicle's center of mass onto the axes of the body-fixed coordinate system; ωx, ωy, ωz are the projections of the UAV's angular velocity vector onto the axes of the body-fixed coordinate system; ξ, η, ζ are the coordinates of the drone's center of mass in the local coordinate system; α, β are the UAV's angle of attack and sideslip angle. X, Y, Z are the projections of the total aerodynamic force acting on the UAV onto the axes of the wind coordinate system: X is drag, Y is lift, Z is side force. g is the acceleration of gravity in the area of the runway, P is the thrust developed by the propeller. Ix, Iy, Iz are the principal moments of inertia of the UAV (the inertia tensor is assumed to be diagonal), q is the dynamic pressure of the oncoming flow, S is the wing area, L is the wingspan, bA is the mean aerodynamic chord; Mx, My, Mz are the dimensionless coefficients of the roll, yaw, and pitch moments, respectively. Algebraic relations are added to the differential equations presented
where v is the UAV's airspeed, Cx, Cy, Cz are the dimensionless coefficients of the aerodynamic forces; δT S is the elevator deflection angle, δdir is the rudder deflection angle, δflap is the flaperon deflection angle (in symmetric mode), δail is the aileron deflection angle (in differential mode).
are the given constant derivatives of the roll, yaw, and pitch moments.
are the given constant coefficients of the aerodynamic forces.
is some given function, which we will assume to be linear in δflap. Let us write down the equations that model the behavior of the elevator, rudder, and ailerons with a lag characterized by a time constant
(15)
The resulting system (1), (15), supplemented by relations (2)—(14) and the control laws for the deflection of the rudders, ailerons, and flaperons, will be complete and closed.
Also, when training multilayer neural networks the most commonly used activation functions are sigmoid ones, named so for their characteristic S-shaped form. Examples of such functions are the hyperbolic tangent (graph (c) in the figure) and the logistic function (graph (d)),
which also visually resemble the inverse trigonometric functions
Graph of the function 
The arcsine of a number x is the value of the angle y, expressed in radians, for which 
The function is continuous and bounded on its entire domain of definition. It is strictly increasing.
for 
for 
(domain of definition),
(range of values).
(the function is odd).
for 
.
for 
for 
Given the function 
On its entire domain of definition it is piecewise monotonic, and therefore the inverse correspondence is not a function. So we consider the interval on which it is strictly increasing and takes all values of its range — 
. Since for the function 
on the interval each value of the function is attained at a single value of the argument, an inverse function 
exists on this interval, and its graph is symmetric to the graph of the function on the interval with respect to the line 
(the graphs of mutually inverse functions are symmetric with respect to the bisector of the first and third quadrants of the coordinate plane 
)
Graph of the function 
The arccosine of a number x is the value of the angle y in radian measure for which 
The function is continuous and bounded on its entire domain of definition. It is strictly decreasing and nonnegative.
for 
for 
(domain of definition),
(range of values).
The function is centrally symmetric about the point 
and is neither even nor odd.
for 
for 
Given the function 
On its entire domain of definition it is piecewise monotonic, and therefore the inverse correspondence is not a function. So we consider the interval on which it is strictly decreasing and takes all of its values — 
On this interval 
is strictly monotonically decreasing and takes each of its values only once, which means that on the interval 
there exists an inverse function 
whose graph is symmetric to the graph of on the interval with respect to the line
Graph of the function 
The arctangent of a number x is the value of the angle 
expressed in radians, for which 
The function 
is defined on the entire real line, is continuous everywhere, and is bounded. It is strictly increasing.
for 
for 
(domain of definition),
(range of values).
(the function is odd).
, for x > 0.
, where 
is the inverse hyperbolic tangent, the hyperbolic area tangent.
Given the function 
On its entire domain of definition it is piecewise monotonic, and therefore the inverse correspondence is not a function (since the single-valuedness requirement is violated). So let us consider the interval on which it is strictly increasing and takes all of its values only once — 
On this interval 
is strictly monotonically increasing and takes all of its values only once; consequently, on the interval 
there exists an inverse function , whose graph is symmetric to the graph of on the interval with respect to the line
Graph of the function 
The arccotangent of a number x is the value of the angle y (in radian measure of angles) for which 
The function 
is defined on the entire real line, is continuous everywhere, and is bounded. It is strictly decreasing and everywhere positive.
for 
for 
The graph of the function is centrally symmetric about the point 
The function is neither even nor odd.
for any 
Given the function 
. On its entire domain of definition it is piecewise monotonic, and therefore the inverse correspondence is not a function. So let us consider the interval on which it is strictly decreasing and takes all of its values only once — 
. On this interval is strictly decreasing and takes all of its values only once; consequently, on the interval there exists an inverse function , whose graph is symmetric to the graph of on the interval with respect to the line
The graph of the arccotangent is obtained from the graph of the arctangent by reflecting the latter about the ordinate axis (that is, replacing the sign of the argument, 
) and shifting it up by π/2; this follows from the formula mentioned above 
Graph of the function 
The arcsecant of a number x is the value of the angle y (in radian measure of angles) for which 
The function is continuous and bounded on its entire domain of definition. It is strictly increasing and everywhere nonnegative.
for 
for 
(domain of definition),
(range of values).
The graph of the function is centrally symmetric about the point The function is neither even nor odd.
for any 
Graph of the function 
The arccosecant of a number x is the value of the angle y (in radian measure of angles) for which 
The function is continuous and bounded on its entire domain of definition. It is strictly decreasing.
for 
for 
(domain of definition),
(range of values).
(the function is odd).
for all 
for all 
for all All inverse trigonometric functions are infinitely differentiable at every point of their domain of definition. The first derivatives:
derivatives of inverse trigonometric functions
Function  |
Derivative  |
Note |
|---|---|---|
 |
 |
Proof The derivative of the arcsine can be found using mutually inverse functions. |
{  |
 |
Proof The derivative of the arccosine can be found using the following identity: |
 |
 |
Proof The derivative of the arctangent can be found using the mutually inverse function: |
 |
 |
Proof The derivative of the arccotangent can be found using the following identity: |
 |
 |
Proof The derivative of the arcsecant can be found using the identity:
Now we differentiate both sides of this identity.
We obtain.
|
 |
 |
Proof The derivative of the arccosecant can be found using this identity: |
For real and complex x:
For real x ≥ 1:
Right triangle ABC
Inverse trigonometric functions are used to compute the angles of a triangle when its sides are known, for example, by means of the law of cosines.
In a right triangle, these functions of the ratios of the sides give the angle directly. Thus, if the leg of length 
is opposite the angle 
, then
To compute the values of inverse trigonometric functions of a complex argument, it is convenient to use formulas that express them in terms of the natural logarithm:
where 
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the triangle's sides are known. Recalling the definitions of sine and cosine in a right triangle, it follows that
Often the hypotenuse is unknown and must be computed before using the arcsine or arccosine, using the Pythagorean theorem: 
where 
is the length of the hypotenuse. In this situation the arctangent can come in handy, since the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet while extending 20 feet horizontally. The roof makes an angle θ with the horizontal, where θ can be computed as follows:
The two-argument function atan2 computes the arctangent of y / x given y and x, but with a range of (−π, π]. In other words, atan2(y, x) is the angle between the positive x-axis of the plane and the point (x, y) on it, with a positive sign for counterclockwise angles (upper half-plane, y > 0) and a negative sign for clockwise angles (lower half-plane, y < 0). It was first introduced in many programming languages, but it is now also widely used in other fields of science and engineering.
In terms of the standard arctan function, that is, with a range of (−π/2, π/2), it can be expressed as follows:
It is also equal to the principal value of the argument of the complex number x + i y.
This function can also be defined using the following tangent half-angle formulas:
provided that either x > 0 or y ≠ 0. However, this fails if x ≤ 0 and y = 0, so the expression is unsuitable for computational use.
The order of arguments (y, x) given above seems to be the most common, and in particular is used in ISO standards such as the C programming language, but some authors may use the opposite convention (x, y), so some care is needed. These variations are detailed at atan2.
In many applications the solution
of the equation 
should be as close as possible to a given value 
. An adequate solution is produced by the modified arctangent function with a parameter
The function 
rounds to the nearest integer.
For angles close to 0 and π, the arccosine is ill-conditioned and will therefore compute the angle with reduced accuracy in a computer implementation (due to the limited number of digits). Similarly, the arcsine is inaccurate for angles close to −π/2 and π/2.
Traditional methods of solving equations with inverse trigonometric functions (arc functions) come down to taking some trigonometric function of both sides, followed by transforming the resulting superpositions using known trigonometric formulas and the formulas given below:
(13)
Formulas (13) are easily derived from the definitions of the arc functions and the basic trigonometric identities. The given formulas can be supplemented with similar ones obtained from the two identities
(14)
and the reduction formulas.
The main drawback of these methods is that the equivalence of the equation is broken during its transformation, so "extraneous" roots can be expected to appear. Identifying extraneous solutions by substituting them into the original equation is often very difficult, either a) because of the complexity of computing non-tabular values of arc functions, or b) because the resulting set of solutions is infinite.
There is a method of solving equations with arc functions in which "extraneous" roots do not arise at all. The method is implemented in the three approaches given below, which differ according to the number of arc functions involved in the equation.
Approach (I): The original equation contains two arc functions. We separate them onto different sides of the equation. We specify the ranges of the left and right sides of the equation by two inequalities. Because arc functions are monotonic, these inequalities are easily solved for the arguments of those functions. The solution of the resulting system of inequalities determines the interval to which the roots of the original equation belong.
Problem 1. Solve the equation
Solution: For comparison, we first use the traditional solution scheme.
Domain of definition:
Next,
Taking the domain of definition into account, 
The resulting interval contains infinitely many "extraneous" solutions, and removing them becomes a separate problem here.
An alternative solution using method (I):
Let 
Since 
and 
the original equation is equivalent to the following system:
Answer:
Problem 2. Solve the equation
Solution: Let 
Let us rewrite the equation in the form:
Since 
the original equation is equivalent to the system:
Problem 3. Solve the equation
Solution: Denote
Since 
we have 
and 
The equation takes the form 
where
and
Since 
is an interval of monotonicity of the tangent, the equation is equivalent to the equation 
When passing to the equation
one may lose those roots for which 
and 
does not exist. In this case that will not happen, since
And the right-hand sides always exist. We obtain the equation
which after transformations takes the form
Since the equation 
has no solutions, what remains is 
Answer: 
Approach (II): Suppose the original equation contains more than two arc functions. In this case the equivalence of the transformations is preserved when the following solution schemes are used:
When solving problems, checking the inequalities 
or 
presents no difficulty and reduces to comparing the ranges of the arc functions that appear in the equation.
Problem 4. Solve the equation: 
Solution: Let 
The original equation is equivalent to the system:
Since 
it suffices to verify that 
The right-hand inequality is true by virtue of the bounds of the arctangent. The left-hand inequality follows from the fact that 
for 
Answer:
Problem 5. Solve the equation: 
Solution: Let 
Then the original equation is equivalent to the system:
(*)
The last inequality obviously follows from the inequalities 
which specify the ranges of the variables. Therefore system (*) is equivalent to the following system: 
The root of the first equation of the system 
is a solution of the original equation. After dividing the first equation by 
we square it.
Problem 6. Solve the equation 
Solution: Let
Since 
both sides of the equation lie in an interval of monotonicity of the sine. Therefore the equation is equivalent to the following:
or
After simplifications we obtain the equation
which has a unique root 
We check and verify that 
is a root of the preceding equation and, consequently, a root of the original equation.
Answer: 
Problem 7. Solve the equation
Solution: Introduce the notation
The given equation takes the form or 
Both sides of the equation lie in the interval If we take the cotangents of both sides of the equation, we can lose only the root corresponding to an angle value equal to 0, since this is the only value in the interval 
at which the cotangent does not exist. Let us check whether the equality 
holds. If 
then 
whence 
and 
For 
we get that 
Thus, 
is a root of the equation.
If 
then we can take the cotangents of both sides of the equation:
This leads to a consequence of the original equation. Expanding the brackets and substituting the expressions of the trigonometric functions 
and 
in terms of 
we obtain the equation
which is equivalent to the system
We obtain two values of the unknown: 
By checking, we verify that both values satisfy the given equation.
Approach (III): To simplify the original equation, in many cases it is convenient to pass from some arc functions to others (for example, from the arcsine or arccosine to the arctangent). Along with formulas (14), the following formulas can be used:
Problem 8. Solve the equation:
Solution: Note that 
does not satisfy the given equation. Therefore, by formulas (15),
So the original equation can be written in the form:
If 
then the equation takes the form:
which is impossible.
If 
then in this case too the equation
has no solutions, because
for 
Answer: no solutions.
Problem 9. Solve the equation 
Solution: 
From the resulting system it follows that 
that is, 
and 
are numbers of the same sign. Indeed, if 
then 
and
If, however, then from the inequalities it immediately follows that 
and 
Consequently, if 
then the equation has no solutions.
If 
then the equation likewise has no solutions, since
Let 
and at least one of the numbers be nonzero. Then we get that
Taking into account the constraints of the system, we find that if and at least one of the numbers is nonzero, then 
If, however, 
and at least one of the numbers is nonzero, then 
Answer: if then the equation has no solutions; if then the equation has no solutions; if and at least one of the numbers is nonzero, then 
if and at least one of the numbers is nonzero, then
Problem 10. Solve the system of equations
Solution: Using the formulas of group 2, we obtain:
Turning to the methods for algebraic systems of equations, we find that 
and 
are the roots of the quadratic equation
We obtain
2.3. Computing the values of inverse trigonometric functions
Example 1. Find 
if 
Solution: Let us estimate 
if
We have
or 
Consequently,
where 
Finally we obtain
Answer: 
Example 2. Prove that if 
then 
Solution: For 
both arcsines exist. For the first this is obvious, and for the second we have
Consequently,
and, all the more,
Introduce the notation
We need to prove that or 
Since 
we have that 
and 
lie in an interval of monotonicity of the sine. Therefore, if we prove that the sines of these arguments are equal, then the equality of the arguments themselves will be proved as well. Since
(the plus sign is taken before the root because 
for 
).
Thus it is proved that 
from which the validity of the equality follows.
Example 3. Prove that the expression 
does not depend on , if 
and simplify it in this case.
Solution: Since 
we have 
Introduce the notation
i.e. 
Consequently,
i.e. the given expression lies in an interval of monotonicity of the sine. Let us find
since 
After substitution we obtain
i.e. 
Answer: 
Example 4. Prove the equality
Solution: Denote the term of the left-hand side by Then we have
Since 
we have 
Example 5. Prove the equality
Solution: Denote the terms of the left-hand side by α, β, γ Then we have
Next, 
Taking into account that these are acute angles, we conclude that 
) we obtain.
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