Inverse trigonometric functions

 ( Trigonometry ) 

The notations , , , etc., as introduced by John Herschel in 1813, are often used as well in English-language sources, much more than the also established , , – conventions consistent with the notation of an inverse function, that is useful (for example) to define the multivalued version of each inverse trigonometric function: $\backslash tan^\{-1\}(x)\; =\; \backslash \{\backslash arctan(x)\; +\; \backslash pi\; k\; \backslash mid\; k\; \backslash in\; \backslash mathbb\; Z\backslash \}\; ~.$ However, this might appear to conflict logically with the common semantics for expressions such as (although only , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal (multiplicative inverse) and inverse function.

The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, . Nevertheless, certain authors advise against using it, since it is ambiguous. Another precarious convention used by a small number of authors is to use an UPPERCASE first letter, along with a “” superscript: , , , etc. Although it is intended to avoid confusion with the reciprocal, which should be represented by , , etc., or, better, by , , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA) use those very same capitalised representations for the standard trig functions, whereas others (Python, SymPy, NumPy, Matlab, MAPLE, etc.) use lower-case.

Hence, since 2009, the standard has specified solely the "arc" prefix for the inverse functions.

For example, using in the sense of multivalued functions, just as the square root function $y\; =\; \backslash sqrt\{x\}$ could be defined from $y^2\; =\; x,$ the function $y\; =\; \backslash arcsin(x)$ is defined so that $\backslash sin(y)\; =\; x.$ For a given real number $x,$ with $-1\; \backslash leq\; x\; \backslash leq\; 1,$ there are multiple (in fact, countably infinitely many) numbers $y$ such that $\backslash sin(y)\; =\; x$; for example, $\backslash sin(0)\; =\; 0,$ but also $\backslash sin(\backslash pi)\; =\; 0,$ $\backslash sin(2\; \backslash pi)\; =\; 0,$ etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each $x$ in the domain, the expression $\backslash arcsin(x)$ will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

The principal inverses are listed in the following table.

• Sine and cosecant begin their period at $2\; \backslash pi\; k-\backslash frac\{\backslash pi\}\{2\}$ (where $k$ is an integer), finish it at $2\; \backslash pi\; k+\backslash frac\{\backslash pi\}\{2\},$ and then reverse themselves over $2\; \backslash pi\; k+\backslash frac\{\backslash pi\}\{2\}$ to $2\; \backslash pi\; k+\backslash frac\{3\backslash pi\}\{2\}.$
• Cosine and secant begin their period at $2\; \backslash pi\; k,$ finish it at $2\; \backslash pi\; k+\backslash pi.$ and then reverse themselves over $2\; \backslash pi\; k+\backslash pi$ to $2\; \backslash pi\; k+2\; \backslash pi.$
• Tangent begins its period at $2\; \backslash pi\; k-\backslash frac\{\backslash pi\}\{2\},$ finishes it at $2\; \backslash pi\; k+\backslash frac\{\backslash pi\}\{2\},$ and then repeats it (forward) over $2\; \backslash pi\; k+\backslash frac\{\backslash pi\}\{2\}$ to $2\; \backslash pi\; k+\backslash frac\{3\; \backslash pi\}\{2\}.$
• Cotangent begins its period at $2\; \backslash pi\; k,$ finishes it at $2\; \backslash pi\; k+\backslash pi,$ and then repeats it (forward) over $2\; \backslash pi\; k+\backslash pi$ to $2\; \backslash pi\; k+2\; \backslash pi.$

This periodicity is reflected in the general inverses, where $k$ is some integer.

The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values $\backslash theta,$ $r,$ $s,$ $x,$ and $y$ all lie within appropriate ranges so that the relevant expressions below are well-defined. Note that "for some $k\; \backslash in\; \backslash Z$" is just another way of saying "for some integer $k.$"

The symbol $\backslash ,\backslash iff\backslash ,$ is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnoteThe expression "LHS $\backslash ,\backslash iff\backslash ,$ RHS" indicates that (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are true, or else (b) the left hand side and right hand side are false; there is option (c) (e.g. it is possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS $\backslash ,\backslash iff\backslash ,$ RHS" would not have been written.

To clarify, suppose that it is written "LHS $\backslash ,\backslash iff\backslash ,$ RHS" where LHS (which abbreviates left hand side) and RHS are both statements that can individually be either be true or false. For example, if $\backslash theta$ and $s$ are some given and fixed numbers and if the following is written: $\backslash tan\; \backslash theta\; =\; s\; \backslash ,\backslash iff\backslash ,\; \backslash theta\; =\; \backslash arctan(s)+\backslash pi\; k\; \backslash quad\; \backslash text\{\; for\; some\; \}\; k\; \backslash in\; \backslash Z$ then LHS is the statement "$\backslash tan\; \backslash theta\; =\; s$". Depending on what specific values $\backslash theta$ and $s$ have, this LHS statement can either be true or false. For instance, LHS is true if $\backslash theta\; =\; 0$ and $s\; =\; 0$ (because in this case $\backslash tan\; \backslash theta\; =\; \backslash tan\; 0\; =\; s$) but LHS is false if $\backslash theta\; =\; 0$ and $s\; =\; 2$ (because in this case $\backslash tan\; \backslash theta\; =\; \backslash tan\; 0\; =\; s$ which is not equal to $s\; =\; 2$); more generally, LHS is false if $\backslash theta\; =\; 0$ and $s\; \backslash neq\; 0.$ Similarly, RHS is the statement "$\backslash theta\; =\; \backslash arctan(s)+\backslash pi\; k$ for some $k\; \backslash in\; \backslash Z$". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values $\backslash theta$ and $s$ have). The logical equality symbol $\backslash ,\backslash iff\backslash ,$ means that (a) if the LHS statement is true then the RHS statement is also true, and moreover (b) if the LHS statement is false then the RHS statement is also false. Similarly, $\backslash ,\backslash iff\backslash ,$ means that (c) if the RHS statement is true then the LHS statement is also true, and moreover (d) if the RHS statement is false then the LHS statement is also false. for more details and an example illustrating this concept).

where the first four solutions can be written in expanded form as:

For example, if $\backslash cos\; \backslash theta\; =\; -1$ then $\backslash theta\; =\; \backslash pi+2\; \backslash pi\; k\; =\; -\backslash pi+2\; \backslash pi\; (1+k)$ for some $k\; \backslash in\; \backslash Z.$ While if $\backslash sin\; \backslash theta\; =\; \backslash pm\; 1$ then $\backslash theta\; =\; \backslash frac\{\backslash pi\}\{2\}+\backslash pi\; k\; =-\backslash frac\{\backslash pi\}\{2\}+\backslash pi\; (k+1)$ for some $k\; \backslash in\; \backslash Z,$ where $k$ will be even if $\backslash sin\; \backslash theta\; =\; 1$ and it will be odd if $\backslash sin\; \backslash theta\; =\; -1.$ The equations $\backslash sec\; \backslash theta\; =\; -1$ and $\backslash csc\; \backslash theta\; =\; \backslash pm\; 1$ have the same solutions as $\backslash cos\; \backslash theta\; =\; -1$ and $\backslash sin\; \backslash theta\; =\; \backslash pm\; 1,$ respectively. In all equations above for those just solved (i.e. except for $\backslash sin$/$\backslash csc\; \backslash theta\; =\; \backslash pm\; 1$ and $\backslash cos$/$\backslash sec\; \backslash theta\; =-1$), the integer $k$ in the solution's formula is uniquely determined by $\backslash theta$ (for fixed $r,\; s,\; x,$ and $y$).

With the help of integer parity it is possible to write a solution to $\backslash cos\; \backslash theta\; =\; x$ that doesn't involve the "plus or minus" $\backslash ,\backslash pm\backslash ,$ symbol:

$cos\; \backslash ;\; \backslash theta\; =\; x\; \backslash quad$ if and only if $\backslash quad\; \backslash theta\; =\; (-1)^h\; \backslash arccos(x)\; +\; \backslash pi\; h\; +\; \backslash pi\; \backslash operatorname\{Parity\}(h)\; \backslash quad$ for some $h\; \backslash in\; \backslash Z.$

$sec\; \backslash ;\; \backslash theta\; =\; r\; \backslash quad$ if and only if $\backslash quad\; \backslash theta\; =\; (-1)^h\; \backslash arcsec(r)\; +\; \backslash pi\; h\; +\; \backslash pi\; \backslash operatorname\{Parity\}(h)\; \backslash quad$ for some $h\; \backslash in\; \backslash Z,$

1. $\backslash ,\backslash theta\; =\; \backslash arccos\; x+2\; \backslash pi\; k\backslash ,$ for some integer $k,$
   or
2. $\backslash ,\backslash theta\; =-\backslash arccos\; x+2\; \backslash pi\; k\backslash ,$ for some integer $k.$

Set of all solutions to elementary trigonometric equations

Thus given a single solution $\backslash theta$ to an elementary trigonometric equation ($\backslash sin\; \backslash theta\; =\; y$ is such an equation, for instance, and because $\backslash sin\; (\backslash arcsin\; y)\; =\; y$ always holds, $\backslash theta\; :=\; \backslash arcsin\; y$ is always a solution), the set of all solutions to it are:

These formulas imply, in particular, that the following hold:

where swapping $\backslash sin\; \backslash leftrightarrow\; \backslash csc,$ swapping $\backslash cos\; \backslash leftrightarrow\; \backslash sec,$ and swapping $\backslash tan\; \backslash leftrightarrow\; \backslash cot$ gives the analogous equations for $\backslash csc,\; \backslash sec,\; \backslash text\{\; and\; \}\; \backslash cot,$ respectively.

So for example, by using the equality $\backslash sin\; \backslash left(\backslash frac\{\backslash pi\}\{2\}-\backslash theta\backslash right)\; =\; \backslash cos\; \backslash theta,$ the equation $\backslash cos\; \backslash theta\; =\; x$ can be transformed into $\backslash sin\; \backslash left(\backslash frac\{\backslash pi\}\{2\}-\backslash theta\backslash right)\; =\; x,$ which allows for the solution to the equation $\backslash ;\backslash sin\; \backslash varphi\; =\; x\backslash ;$ (where $\backslash varphi\; :=\; \backslash frac\{\backslash pi\}\{2\}-\backslash theta$) to be used; that solution being: $\backslash varphi\; =\; (-1)^k\; \backslash arcsin\; (x)+\backslash pi\; k\; \backslash ;\; \backslash text\{\; for\; some\; \}\; k\; \backslash in\; \backslash Z,$ which becomes: $$\backslash frac\{\backslash pi\}\{2\}-\backslash theta\; ~=~\; (-1)^k\; \backslash arcsin\; (x)+\backslash pi\; k\; \backslash quad\; \backslash text\{\; for\; some\; \}\; k\; \backslash in\; \backslash Z$$ where using the fact that $(-1)^\{k\}\; =\; (-1)^\{-k\}$ and substituting $h\; :=-k$ proves that another solution to $\backslash ;\backslash cos\; \backslash theta\; =\; x\backslash ;$ is: $$\backslash theta\; ~=~\; (-1)^\{h+1\}\; \backslash arcsin\; (x)+\backslash pi\; h+\backslash frac\{\backslash pi\}\{2\}\; \backslash quad\; \backslash text\{\; for\; some\; \}\; h\; \backslash in\; \backslash Z.$$ The substitution $\backslash ;\backslash arcsin\; x\; =\; \backslash frac\{\backslash pi\}\{2\}-\backslash arccos\; x\backslash ;$ may be used express the right hand side of the above formula in terms of $\backslash ;\backslash arccos\; x\backslash ;$ instead of $\backslash ;\backslash arcsin\; x.\backslash ;$

$\backslash begin\{align\}$

Negative arguments:

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Reciprocal arguments:

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Useful identities if one only has a fragment of a sine table:

$\backslash begin\{align\}$

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).

A useful form that follows directly from the table above is

$\backslash arctan(x)\; =\; \backslash arccos\backslash left(\backslash sqrt\{\backslash frac\{1\}\{1+x^2\}\}\backslash right)\backslash ,\; ,\; \backslash text\{\; if\; \}\; x\backslash geq\; 0$.

It is obtained by recognizing that $\backslash cos\backslash left(\backslash arctan\backslash left(x\backslash right)\backslash right)\; =\; \backslash sqrt\{\backslash frac\{1\}\{1+x^2\}\}\; =\; \backslash cos\backslash left(\backslash arccos\backslash left(\backslash sqrt\{\backslash frac\{1\}\{1+x^2\}\}\backslash right)\backslash right)$.

From the half-angle formula, $\backslash tan\backslash left(\backslash tfrac\{\backslash theta\}\{2\}\backslash right)\; =\; \backslash tfrac\{\backslash sin(\backslash theta)\}\{1\; +\; \backslash cos(\backslash theta)\}$, we get:

$\backslash begin\{align\}$

$\backslash arctan(u)\; \backslash pm\; \backslash arctan(v)\; =\; \backslash arctan\backslash left(\backslash frac\{u\; \backslash pm\; v\}\{1\; \backslash mp\; uv\}\backslash right)\; \backslash pmod\; \backslash pi\; \backslash ,\; ,\; \backslash quad\; u\; v\; \backslash ne\; 1\; \backslash ,\; .$

$\backslash tan(\backslash alpha\; \backslash pm\; \backslash beta)\; =\; \backslash frac\{\backslash tan(\backslash alpha)\; \backslash pm\; \backslash tan(\backslash beta)\}\{1\; \backslash mp\; \backslash tan(\backslash alpha)\backslash tan(\backslash beta)\}\; \backslash ,\; ,$

$\backslash alpha\; =\; \backslash arctan(u)\; \backslash ,\; ,\; \backslash quad\; \backslash beta\; =\; \backslash arctan(v)\; \backslash ,\; .$

$\backslash begin\{align\}$

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& a & & b & & c && -i\ln\left(\frac{a+ib}{c}\right) && \theta && \theta_{a,b\in\R}\\
     

 \sin(\phi) &= z \\
       \phi &= \arcsin(z)
     

$\backslash theta\; =\; \backslash arcsin\; \backslash left(\; \backslash frac\{\backslash text\{opposite\}\}\{\backslash text\{hypotenuse\}\}\; \backslash right)\; =\; \backslash arccos\; \backslash left(\; \backslash frac\{\backslash text\{adjacent\}\}\{\backslash text\{hypotenuse\}\}\; \backslash right)\; .$

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: $a^2+b^2=h^2$ where $h$ is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.

$\backslash theta\; =\; \backslash arctan\; \backslash left(\; \backslash frac\{\backslash text\{opposite\}\}\{\backslash text\{adjacent\}\}\; \backslash right)\; \backslash ,\; .$

For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:

$\backslash theta$

In terms of the standard arctan function, that is with range of , it can be expressed as follows:

$$\backslash operatorname\{atan2\}(y,\; x)\; =\; \backslash begin\{cases\}$$

 \arctan\left(\frac y x\right)       & \quad x > 0 \\
 \arctan\left(\frac y x\right) + \pi & \quad y \ge 0,\; x < 0 \\
 \arctan\left(\frac y x\right) - \pi & \quad y < 0,\; x < 0 \\
  \frac{\pi}{2}   & \quad y > 0,\; x = 0 \\
 -\frac{\pi}{2}   & \quad y < 0,\; x = 0 \\
 \text{undefined} & \quad y = 0,\; x = 0
     

It also equals the principal value of the argument of the complex number .

This limited version of the function above may also be defined using the tangent half-angle formulae as follows: $$\backslash operatorname\{atan2\}(y,\; x)\; =\; 2\backslash arctan\backslash left(\backslash frac\{y\}\{\backslash sqrt\{x^2\; +\; y^2\}\; +\; x\}\backslash right)$$ provided that either or . However this fails if given and so the expression is unsuitable for computational use.

The above argument order (, ) seems to be the most common, and in particular is used in such as the C programming language, but a few authors may use the opposite convention (, ) so some caution is warranted.

The function $\backslash operatorname\{rni\}$ rounds to the nearest integer.