Difference between revisions of "Trigonometric Equations"

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[[File:TrigFunctionDiagram.svg|thumb|Plot of the six trigonometric functions, the unit circle, and a line for the angle <math>\theta = 0.7</math> radians. The points labelled {{color|#A00|1}}, {{color|#00A|Sec(θ)}}, {{color|#0A0|Csc(θ)}} represent the length of the line segment from the origin to that point. {{color|#A00|Sin(θ)}}, {{color|#00A|Tan(θ)}}, and {{color|#0A0|1}} are the heights to the line starting from the <math>x</math>-axis, while {{color|#A00|Cos(θ)}}, {{color|#00A|1}}, and {{color|#0A0|Cot(θ)}} are lengths along the <math>x</math>-axis starting from the origin.]]
 
  
 
The functions sine, cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions. Their usual abbreviations are <math>\sin (\theta), \cos (\theta),</math> and <math>\tan (\theta),</math> respectively, where <math>\theta</math> denotes the angle. The parentheses around the argument of the functions are often omitted, e.g., <math>\sin \theta</math> and <math>\cos \theta,</math> if an interpretation is unambiguously possible.
 
The functions sine, cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions. Their usual abbreviations are <math>\sin (\theta), \cos (\theta),</math> and <math>\tan (\theta),</math> respectively, where <math>\theta</math> denotes the angle. The parentheses around the argument of the functions are often omitted, e.g., <math>\sin \theta</math> and <math>\cos \theta,</math> if an interpretation is unambiguously possible.
  
The sine of an angle is defined, in the context of a [[right triangle]], as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle (the hypotenuse).
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The sine of an angle is defined, in the context of a right triangle, as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle (the hypotenuse).
 
:<math>\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}.</math>
 
:<math>\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}.</math>
 
The cosine of an angle in this context is the ratio of the length of the side that is adjacent to the angle divided by the length of the hypotenuse.
 
The cosine of an angle in this context is the ratio of the length of the side that is adjacent to the angle divided by the length of the hypotenuse.
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\\
 
\\
 
\end{align}
 
\end{align}
</math><ref>{{harvnb|Abramowitz|Stegun|1972|loc=p. 73, 4.3.45}}</ref>
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</math>
  
Taking the [[multiplicative inverse]] of both sides of the each equation above results in the equations for <math>\csc = \frac{1}{\sin}, \;\sec = \frac{1}{\cos}, \text{ and } \cot = \frac{1}{\tan}.</math>  
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Taking the multiplicative inverse of both sides of the each equation above results in the equations for <math>\csc = \frac{1}{\sin}, \;\sec = \frac{1}{\cos}, \text{ and } \cot = \frac{1}{\tan}.</math>  
 
The right hand side of the formula above will always be flipped.  
 
The right hand side of the formula above will always be flipped.  
 
For example, the equation for <math>\cot(\arcsin x)</math> is:
 
For example, the equation for <math>\cot(\arcsin x)</math> is:
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The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.  
 
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 <math>\theta, r, s, x,</math> and <math>y</math> all lie within appropriate ranges so that the relevant expressions below are [[well-defined]].  
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It is assumed that the given values <math>\theta, r, s, x,</math> and <math>y</math> all lie within appropriate ranges so that the relevant expressions below are well-defined.  
In the table below, "for some {{nowrap|<math>k \in \Z</math>}}" is just another way of saying "for some [[integer]] <math>k.</math>"
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In the table below, "for some <math>k \in \Z</math>" is just another way of saying "for some integer <math>k.</math>"
  
 
{| class="wikitable" style="border: none;"
 
{| class="wikitable" style="border: none;"
 
|+  
 
|+  
 
|-
 
|-
! Equation !! [[if and only if]] !! colspan="6"|Solution !! where...
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! Equation !! if and only if !! colspan="6"|Solution !! where...
 
|-
 
|-
 
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;"|<math>\sin \theta = y</math>
 
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;"|<math>\sin \theta = y</math>
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|}
 
|}
  
For example, if <math>\cos \theta = -1</math> then <math>\theta = \pi + 2 \pi k = -\pi + 2 \pi (1 + k)</math> for some <math>k \in \Z.</math> While if <math>\sin \theta = \pm 1</math> then <math>\theta = \frac{\pi}{2} + \pi k = - \frac{\pi}{2} + \pi (k + 1)</math> for some <math>k \in \Z,</math> where <math>k</math> is even if <math>\sin \theta = 1</math>; odd if <math>\sin \theta = -1.</math> The equations <math>\sec \theta = -1</math> and <math>\csc \theta = \pm 1</math> have the same solutions as <math>\cos \theta = -1</math> and <math>\sin \theta = \pm 1,</math> respectively. In all equations above {{em|except}} for those just solved (i.e. except for <math>\sin</math>/<math>\csc \theta = \pm 1</math> and <math>\cos</math>/<math>\sec \theta = - 1</math>), for fixed <math>r, s, x,</math> and <math>y,</math> the integer <math>k</math> in the solution's formula is uniquely determined by <math>\theta.</math>  
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For example, if <math>\cos \theta = -1</math> then <math>\theta = \pi + 2 \pi k = -\pi + 2 \pi (1 + k)</math> for some <math>k \in \Z.</math> While if <math>\sin \theta = \pm 1</math> then <math>\theta = \frac{\pi}{2} + \pi k = - \frac{\pi}{2} + \pi (k + 1)</math> for some <math>k \in \Z,</math> where <math>k</math> is even if <math>\sin \theta = 1</math>; odd if <math>\sin \theta = -1.</math> The equations <math>\sec \theta = -1</math> and <math>\csc \theta = \pm 1</math> have the same solutions as <math>\cos \theta = -1</math> and <math>\sin \theta = \pm 1,</math> respectively. In all equations above except for those just solved (i.e. except for <math>\sin</math>/<math>\csc \theta = \pm 1</math> and <math>\cos</math>/<math>\sec \theta = - 1</math>), for fixed <math>r, s, x,</math> and <math>y,</math> the integer <math>k</math> in the solution's formula is uniquely determined by <math>\theta.</math>
 
 
{{EqualOrNegativeIdenticalTrigonometricFunctionsSolutions|includeTableDescription=true|style=}}
 
  
 
=== Other identities ===
 
=== Other identities ===
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which hold whenever they are well-defined (that is, whenever <math>x, r, s, -x, -r, \text{ and } -s</math> are in the domains of the relevant functions).  
 
which hold whenever they are well-defined (that is, whenever <math>x, r, s, -x, -r, \text{ and } -s</math> are in the domains of the relevant functions).  
  
Aslo,<ref name=Wu>Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", ''Mathematics Magazine'' 77(3), June 2004, p. 189.</ref>
 
 
<math display=block>
 
<math display=block>
 
\begin{align}
 
\begin{align}
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==Resources==
 
==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Equations/Esparza%201093%20Notes%203.3B.pdf Trigonometric Equations]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Equations/Esparza%201093%20Notes%203.3B.pdf Trigonometric Equations]. Written notes created by Professor Esparza, UTSA.
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 +
== Licensing ==
 +
Content obtained and/or adapted from:
 +
* [https://en.wikipedia.org/wiki/List_of_trigonometric_identities List of trigonometric identities, Wikipedia] under a CC BY-SA license

Latest revision as of 16:52, 28 October 2021

The functions sine, cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions. Their usual abbreviations are and respectively, where denotes the angle. The parentheses around the argument of the functions are often omitted, e.g., and if an interpretation is unambiguously possible.

The sine of an angle is defined, in the context of a right triangle, as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle (the hypotenuse).

The cosine of an angle in this context is the ratio of the length of the side that is adjacent to the angle divided by the length of the hypotenuse.

The tangent of an angle in this context is the ratio of the length of the side that is opposite to the angle divided by the length of the side that is adjacent to the angle. This is the same as the ratio of the sine to the cosine of this angle, as can be seen by substituting the definitions of and from above:

The remaining trigonometric functions secant (), cosecant (), and cotangent () are defined as the reciprocal functions of cosine, sine, and tangent, respectively. Rarely, these are called the secondary trigonometric functions:

These definitions are sometimes referred to as ratio identities.

Inverse trigonometric functions

The inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the inverse function for the sine, known as the inverse sine () or arcsine (arcsin or asin), satisfies

and

This article will denote the inverse of a trigonometric function by prefixing its name with "". The notation is shown in the table below.

Compositions of trig and inverse trig functions

Taking the multiplicative inverse of both sides of the each equation above results in the equations for The right hand side of the formula above will always be flipped. For example, the equation for is:

while the equations for and are:

Solutions to elementary trigonometric equations

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 and all lie within appropriate ranges so that the relevant expressions below are well-defined. In the table below, "for some " is just another way of saying "for some integer "

Equation if and only if Solution where...
for some
for some
for some
for some
for some
for some

For example, if then for some While if then for some where is even if ; odd if The equations and have the same solutions as and respectively. In all equations above except for those just solved (i.e. except for / and /), for fixed and the integer in the solution's formula is uniquely determined by

Other identities

These inverse trigonometric functions are related to one another by the formulas:

which hold whenever they are well-defined (that is, whenever are in the domains of the relevant functions).

Resources

Licensing

Content obtained and/or adapted from: