Difference between revisions of "Trigonometric equations involving a single trig function"

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'''Trigonometric equations''' are equations including trigonometric functions. If they have only such functions and constants, then the solution involves finding an unknown which is an argument to a trigonometric function.
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==Basic trigonometric equations==
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===sin(''x'') = n===
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<div style="padding-left: 1em; float: right">
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{| border="1" cellspacing="0" cellpadding="5" style="text-align: center"
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|colspan="2"| [[Image:Sin unit circle.svg|300px]]
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|-
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! <math>n</math> !! <math>\sin(x)=n</math>
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|-
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| <math>|n|<1</math>
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| <math>\begin{matrix}x=\alpha+2k\pi \\
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x=\pi-\alpha+2k\pi \\
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\alpha\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\end{matrix}</math>
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|-
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| <math>n=-1</math>
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| <math>x=-\begin{matrix}\frac{\pi}{2}\end{matrix}+2k\pi</math>
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|-
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| <math>n=0</math>
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| <math>x=k\pi</math>
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|-
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| <math>n=1</math>
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| <math>x=\begin{matrix}\frac{\pi}{2}\end{matrix}+2k\pi</math>
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|-
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| <math>|n|>1</math>
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| <math>x\in\varnothing</math>
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|}
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</div>
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 +
The equation <math>\sin(x)=n</math> has solutions only when <math>n</math> is within the interval <math>[-1,1]</math> . If <math>n</math> is within this interval, then we first find an <math>\alpha</math> such that:
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:<math>\alpha=\arcsin(n)</math>
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The solutions are then:
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:<math>x=\alpha+2k\pi</math>
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:<math>x=\pi-\alpha+2k\pi</math>
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Where <math>k</math> is an integer.
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 +
In the cases when <math>n</math> equals 1, 0 or -1 these solutions have simpler forms which are summarized in the table on the right.
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 +
For example, to solve:
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:<math>\sin\bigl(\tfrac{x}{2}\bigr)=\frac{\sqrt3}{2}</math>
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First find <math>\alpha</math> :
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:<math>\alpha=\arcsin\bigl(\tfrac{\sqrt3}{2}\bigr)=\frac{\pi}{3}</math>
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Then substitute in the formulae above:
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:<math>\frac{x}{2}=\frac{\pi}{3}+2k\pi</math>
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:<math>\frac{x}{2}=\pi-\frac{\pi}{3}+2k\pi</math>
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Solving these linear equations for <math>x</math> gives the final answer:
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:<math>x=\frac{2\pi}{3}(1+6k)</math>
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:<math>x=\frac{4\pi}{3}(1+3k)</math>
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Where <math>k</math> is an integer.
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<div style="clear: both"></div>
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===cos(''x'') = n===
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<div style="padding-left: 1em; float: right">
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{| border="1" cellspacing="0" cellpadding="5" style="text-align: center"
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|colspan="2"| [[Image:Cos unit circle.svg]]
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|-
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! <math>n</math> !! <math>\cos(x)=n</math>
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|-
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| <math>|n|<1</math>
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| <math>\begin{matrix}x=\pm\alpha+2k\pi \\
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\alpha\in[0,\pi]\end{matrix}</math>
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|-
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| <math>n=-1</math>
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| <math>x=\pi+2k\pi</math>
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|-
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| <math>n=0</math>
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| <math>x=\begin{matrix}\frac{\pi}{2}\end{matrix}+k\pi</math>
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|-
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| <math>n=1</math>
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| <math>x=2k\pi</math>
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|-
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| <math>|n|>1</math>
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| <math>x\in\varnothing</math>
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|}
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</div>
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 +
Like the sine equation, an equation of the form <math>\cos(x)=n</math> only has solutions when n is in the interval <math>[-1,1]</math> . To solve such an equation we first find one angle <math>\alpha</math> such that:
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:<math>\alpha=\arccos(n)</math>
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Then the solutions for <math>x</math> are:
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:<math>x=\pm\alpha+ 2k\pi</math>
 +
Where <math>k</math> is an integer.
 +
 +
Simpler cases with <math>n</math> equal to 1, 0 or -1 are summarized in the table on the right.
 +
<div style="clear: both"></div>
 +
 +
===tan(''x'') = n===
 +
<div style="padding-left: 1em; float: right">
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{| border="1" cellspacing="0" cellpadding="5" style="text-align: center"
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|colspan="2"| [[Image:Tan unit circle.svg]]
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|-
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! <math>n!</math> !! <math>\tan(x)=n</math>
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|-
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| General<br/>case
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| <math>\begin{matrix}x=\alpha+k\pi \\
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\alpha\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\end{matrix}</math>
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|-
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| <math>n=-1</math>
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| <math>x=-\begin{matrix}\frac{\pi}{4}\end{matrix}+k\pi</math>
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|-
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| <math>n=0</math>
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| <math>x=k\pi</math>
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|-
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| <math>n=1</math>
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| <math>x=\begin{matrix}\frac{\pi}{4}\end{matrix}+k\pi</math>
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|}
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</div>
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An equation of the form <math>\tan(x)=n</math> has solutions for any real <math>n</math> . To find them we must first find an angle <math>\alpha</math> such that:
 +
:<math>\alpha=\arctan(n)</math>
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After finding <math>\alpha</math> , the solutions for <math>x</math> are:
 +
:<math>x=\alpha+k\pi</math>
 +
When <math>n</math> equals 1, 0 or -1 the solutions have simpler forms which are shown in the table on the right.
 +
<div style="clear: both"></div>
 +
 +
===cot(''x'') = n===
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<div style="padding-left: 1em; float: right">
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{| border="1" cellspacing="0" cellpadding="5" style="text-align: center"
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|colspan="2"| [[Image:Cot unit circle.svg]]
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|-
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! <math>n</math> !! <math>\cot(x)=n</math>
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|-
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| General<br/>case
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| <math>\begin{matrix}x=\alpha+k\pi \\
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\alpha\in\left[0;\pi\right]\end{matrix}</math>
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|-
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| <math>n=-1</math>
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| <math>x=-\begin{matrix}\frac{3\pi}{4}\end{matrix}+k\pi</math>
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|-
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| <math>n=0</math>
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| <math>x=\begin{matrix}\frac{\pi}{2}\end{matrix}+k\pi</math>
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|-
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| <math>n=1</math>
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| <math>x=\begin{matrix}\frac{\pi}{4}\end{matrix}+k\pi</math>
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|}
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</div>
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 +
The equation <math>\cot(x)=n</math> has solutions for any real <math>n</math> . To find them we must first find an angle <math>\alpha</math> such that:
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:<math>\alpha=\arccot(n)</math>
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After finding <math>\alpha</math> , the solutions for <math>x</math> are:
 +
:<math>x=\alpha+k\pi</math>
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When <math>n</math> equals 1, 0 or -1 the solutions have simpler forms which are shown in the table on the right.
 +
<div style="clear: both"></div>
 +
 +
===csc(''x'') = n and sec(''x'') = n===
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The trigonometric equations <math>\csc(x)=n</math> and <math>\sec(x)=n</math> can be solved by transforming them to other basic equations:
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:<math>\csc(x)=n\ \Leftrightarrow\ \frac{1}{\sin(x)}=n\ \Leftrightarrow\ \sin(x)=\frac{1}{n}</math>
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:<math>\sec(x)=n\ \Leftrightarrow\ \frac{1}{\cos(x)}=n\ \Leftrightarrow\ \cos(x)=\frac{1}{n}</math>
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==Further examples==
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Generally, to solve trigonometric equations we must first transform them to a basic trigonometric equation using the [[../Trigonometric_Identities_Reference|trigonometric identities]]. This sections lists some common examples.
 +
 +
===''a'' sin(''x'')+''b'' cos(''x'') = ''c''===
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To solve this equation we will use the identity:
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:<math>a\sin(x)+b\cos(x)=\sqrt{a^2+b^2}\sin(x+\alpha)</math>
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:<math>\alpha=\begin{cases}\arctan\bigl(\frac{b}{a}\bigr), & \mbox{if } a>0 \\ \pi+\arctan\bigl(\frac{b}{a}\bigr), & \mbox{if } a<0 \end{cases}</math>
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The equation becomes:
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:<math>\sqrt{a^2+b^2}\sin(x+\alpha)=c</math>
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:<math>\sin(x+\alpha)=\frac{c}{\sqrt{a^2+b^2}}</math>
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 +
This equation is of the form <math>\sin(x)=n</math> and can be solved with the formulae given above.
 +
 +
For example we will solve:
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:<math>\sin(3x)-\sqrt3\cos(3x)=-\sqrt3</math>
 +
In this case we have:
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:<math>a=1,b=-\sqrt3</math>
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:<math>\sqrt{a^2+b^2}=\sqrt{1^2+\Big(-\sqrt3\Big)^2}=2</math>
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:<math>\alpha=\arctan\Big(-\sqrt3\Big)=-\frac{\pi}{3}</math>
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Apply the identity:
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:<math>2\sin\left(3x-\frac{\pi}{3}\right)=-\sqrt3</math>
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:<math>\sin\left(3x-\frac{\pi}{3}\right)=-\frac{\sqrt3}{2}</math>
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So using the formulae for <math>\sin(x)=n</math> the solutions to the equation are:
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:<math>3x-\frac{\pi}{3}=-\frac{\pi}{3}+2k\pi\ \Leftrightarrow\ x=\frac{2k\pi}{3}</math>
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:<math>3x-\frac{\pi}{3}=\pi+\frac{\pi}{3}+2k\pi\ \Leftrightarrow\ x=\frac{\pi}{9}(6k+5)</math>
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Where <math>k</math> is an integer.
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 +
==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20equations%20involving%20a%20single%20trig%20function/Esparza%201093%20Notes%203.3A.pdf Trigonometric equations involving a single trig function]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20equations%20involving%20a%20single%20trig%20function/Esparza%201093%20Notes%203.3A.pdf Trigonometric equations involving a single trig function]. Written notes created by Professor Esparza, UTSA.
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20equations%20involving%20a%20single%20trig%20function/Esparza%201093%20Notes%203.3ACont.pdf Trigonometric equations involving a single trig function]. Written notes created by Professor Esparza, UTSA.
+
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20equations%20involving%20a%20single%20trig%20function/Esparza%201093%20Notes%203.3ACont.pdf Trigonometric equations involving a single trig function continued]. Written notes created by Professor Esparza, UTSA.
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== Licensing ==
 +
Content obtained and/or adapted from:
 +
* [https://en.wikibooks.org/wiki/Trigonometry/Solving_Trigonometric_Equations Solving Trigonometric Equations, Wikibooks: Trigonometry] under a CC BY-SA license

Latest revision as of 16:48, 28 October 2021

Trigonometric equations are equations including trigonometric functions. If they have only such functions and constants, then the solution involves finding an unknown which is an argument to a trigonometric function.

Basic trigonometric equations

sin(x) = n

Sin unit circle.svg

The equation has solutions only when is within the interval . If is within this interval, then we first find an such that:

The solutions are then:

Where is an integer.

In the cases when equals 1, 0 or -1 these solutions have simpler forms which are summarized in the table on the right.

For example, to solve:

First find  :

Then substitute in the formulae above:

Solving these linear equations for gives the final answer:

Where is an integer.

cos(x) = n

Cos unit circle.svg

Like the sine equation, an equation of the form only has solutions when n is in the interval . To solve such an equation we first find one angle such that:

Then the solutions for are:

Where is an integer.

Simpler cases with equal to 1, 0 or -1 are summarized in the table on the right.

tan(x) = n

Tan unit circle.svg
General
case

An equation of the form has solutions for any real . To find them we must first find an angle such that:

After finding , the solutions for are:

When equals 1, 0 or -1 the solutions have simpler forms which are shown in the table on the right.

cot(x) = n

Cot unit circle.svg
General
case

The equation has solutions for any real . To find them we must first find an angle such that:

After finding , the solutions for are:

When equals 1, 0 or -1 the solutions have simpler forms which are shown in the table on the right.

csc(x) = n and sec(x) = n

The trigonometric equations and can be solved by transforming them to other basic equations:

Further examples

Generally, to solve trigonometric equations we must first transform them to a basic trigonometric equation using the [[../Trigonometric_Identities_Reference|trigonometric identities]]. This sections lists some common examples.

a sin(x)+b cos(x) = c

To solve this equation we will use the identity:

The equation becomes:

This equation is of the form and can be solved with the formulae given above.

For example we will solve:

In this case we have:

Apply the identity:

So using the formulae for the solutions to the equation are:

Where is an integer.

Resources

Licensing

Content obtained and/or adapted from: