Trigonometric Equations - Revision history

Khanh at 22:52, 28 October 2021

2021-10-28T22:52:53Z

Lila at 21:07, 12 October 2021

2021-10-12T21:07:54Z

Lila at 21:06, 12 October 2021

2021-10-12T21:06:37Z

Lila: /* Compositions of trig and inverse trig functions */

2021-10-12T21:05:49Z

Compositions of trig and inverse trig functions

Lila: /* Solutions to elementary trigonometric equations */

2021-10-12T21:04:35Z

Solutions to elementary trigonometric equations

Lila: /* Solutions to elementary trigonometric equations */

2021-10-12T21:03:02Z

Solutions to elementary trigonometric equations

Lila at 21:01, 12 October 2021

2021-10-12T21:01:39Z

Lila at 20:58, 12 October 2021

2021-10-12T20:58:39Z

Lila at 20:57, 12 October 2021

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Lila at 20:55, 12 October 2021

2021-10-12T20:55:07Z

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← Older revision		Revision as of 22:52, 28 October 2021
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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.
		+
		+	== 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

← Older revision		Revision as of 21:07, 12 October 2021
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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.

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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).
 <math display=block>
 \begin{align}

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 \\
 \end{align}
-</math><ref>{{harvnb|Abramowitz|Stegun|1972|loc=p. 73, 4.3.45}}</ref>
+</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>
+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.
 For example, the equation for <math>\cot(\arcsin x)</math> is:

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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>
+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>
 === Other identities ===

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 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]].
+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 <math>k \in \Z</math>" is just another way of saying "for some integer <math>k.</math>"
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 |+
 |-
-! Equation !! [[if and only if]] !! colspan="6"|Solution !! where...
+! 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>

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 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).
+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>
 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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 [[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 [[Trigonometric functions#tangent|tangent]] of an angle are sometimes referred to as the {{em|primary}} or {{em|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]]).
+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>
 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.
 :<math>\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}.</math>
-The [[Tangent function|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 <math>\sin</math> and <math>\cos</math> from above:
+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 <math>\sin</math> and <math>\cos</math> from above:
 :<math>\tan\theta = \frac{\sin\theta}{\cos\theta}=\frac{\text{opposite}}{\text{adjacent}}.</math>
-The remaining trigonometric functions secant (<math>\sec</math>), cosecant (<math>\csc</math>), and cotangent (<math>\cot</math>) are defined as the [[Trigonometric functions#Cosecant, secant, and cotangent|reciprocal functions]] of cosine, sine, and tangent, respectively. Rarely, these are called the secondary trigonometric functions:
+The remaining trigonometric functions secant (<math>\sec</math>), cosecant (<math>\csc</math>), and cotangent (<math>\cot</math>) are defined as the reciprocal functions of cosine, sine, and tangent, respectively. Rarely, these are called the secondary trigonometric functions:
 :<math>\sec\theta = \frac{1}{\cos\theta},\quad\csc\theta = \frac{1}{\sin\theta},\quad\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}.</math>
-These definitions are sometimes referred to as [[Proofs of trigonometric identities#Ratio identities|ratio identities]].
+These definitions are sometimes referred to as ratio identities.
 == Inverse trigonometric functions ==
-{{main|Inverse trigonometric functions}}
-The inverse trigonometric functions are partial [[inverse function]]s for the trigonometric functions. For example, the inverse function for the sine, known as the '''inverse sine''' (<math>\sin^{-1}</math>) or '''arcsine''' ({{math|arcsin}} or {{math|asin}}), satisfies
+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''' (<math>\sin^{-1}</math>) or '''arcsine''' ({{math|arcsin}} or {{math|asin}}), satisfies
 <math display=block>\sin(\arcsin x) = x\quad\text{for} \quad |x| \leq 1</math>
 and
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 This article will denote the inverse of a trigonometric function by prefixing its name with "<math>\operatorname{arc}</math>". The notation is shown in the table below.
 === Compositions of trig and inverse trig functions ===