4.3 Trigonometric relations - Revision history

Jdinglis at 20:17, 10 November 2008

2008-11-10T20:17:08Z

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	<br>		<br>
-	When an angle <math>v</math> is reflected in the ''x''-axis it becomes<math>-v</math>.	+	When an angle <math>v</math> is reflected in the ''x''-axis it becomes <math>-v</math>.

Jdinglis at 20:16, 10 November 2008

2008-11-10T20:16:42Z

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	</div>		</div>

-	Instead of trying to learn all of these relationships by heart, it is be better to learn how to derive them from the unit circle.	+	Instead of trying to learn all of these relationships by heart, it is better to learn how to derive them from the unit circle.

Jdinglis at 20:16, 10 November 2008

2008-11-10T20:16:18Z

←Older revision		Revision as of 20:16, 10 November 2008
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	== Symmetries ==		== Symmetries ==

-	With the help of the unit circle by exploiting the symmetries ~~of the trigonometric functions one obtains~~ a large number of relationships between the cosine and sine functions:	+	With the help of the unit circle and by exploiting the symmetries we obtain a large number of relationships between the cosine and sine functions:

	<div class="regel">		<div class="regel">

Jdinglis at 19:21, 10 November 2008

2008-11-10T19:21:52Z

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	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\cos v = 1 – \sin^2\!\left(\frac{v}{2}\right) – \sin^2\!\left(\frac{v}{2}\right)		\cos v = 1 – \sin^2\!\left(\frac{v}{2}\right) – \sin^2\!\left(\frac{v}{2}\right)
-	= 1 – 2\sin^2\!\left(frac{v}{2}\right)</math>}}	+	= 1 – 2\sin^2\!\left(\frac{v}{2}\right)</math>}}
	so that		so that
	<div class="regel">		<div class="regel">
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-	'''Keep in mind that:'''	+	'''Keep in mind that...'''

-	The unit circle is an invaluable tool for ~~finding~~ trigonometric relationships. ~~They~~ are ~~a multitude~~ and there is no point in trying to learn all of them by heart. It is also time-consuming to have to look them up all the time. Therefore, it is much better that you learn how to use the unit circle.	+	The unit circle is an invaluable tool for deriving trigonometric relationships. There are lots of these, and there is no point in trying to learn all of them by heart. It is also time-consuming to have to look them up all the time. Therefore, it is much better that you learn how to use the unit circle.

-	The most famous trigonometric formula is the so-called Pythagorean identity. It applies to all angles, not just ~~for~~ acute angles. It is based on the Pythagoras theorem.	+	The most famous trigonometric formula is the so-called Pythagorean identity. It applies to all angles, not just acute angles. It is based on the Pythagoras theorem.

Jdinglis at 19:19, 10 November 2008

2008-11-10T19:19:39Z

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	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\cos v = 1 – \sin^2\!\left(\frac{v}{2}\right) – \sin^2\!\left(\frac{v}{2}\right)		\cos v = 1 – \sin^2\!\left(\frac{v}{2}\right) – \sin^2\!\left(\frac{v}{2}\right)
-	= 1 – 2\sin^2\!\\left(frac{v}{2}\right)</math>}}	+	= 1 – 2\sin^2\!\left(frac{v}{2}\right)</math>}}
	so that		so that
	<div class="regel">		<div class="regel">

Jdinglis at 19:17, 10 November 2008

2008-11-10T19:17:39Z

←Older revision		Revision as of 19:17, 10 November 2008
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	From these relationships, we can also derive formulas for half angles. By replacing <math>2v</math> by <math>v</math>, and consequently <math>v</math> by <math>v/2</math>, in the formula for <math>\cos 2v</math> we get that		From these relationships, we can also derive formulas for half angles. By replacing <math>2v</math> by <math>v</math>, and consequently <math>v</math> by <math>v/2</math>, in the formula for <math>\cos 2v</math> we get that
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
-	\cos v = \cos^2\!\frac{v}{2}\ –\ \sin^2\!\frac{v}{2}\,\mbox{.}</math>}}	+	\cos v = \cos^2\!\left(\frac{v}{2}\right)\ –\ \sin^2\!\left(\frac{v}{2}\right)\,\mbox{.}</math>}}

-	If we want a formula for <math>\sin(v/2)</math> we use the Pythagorean identity to write <math>\cos^2(v/2)<\math> in terms of <math>\sin^2(v/2)</math>. Then	+	If we want a formula for <math>\sin(v/2)</math> we use the Pythagorean identity to write <math>\cos^2(v/2)</math> in terms of <math>\sin^2(v/2)</math>. Then
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
-	\cos v = 1 – \sin^2\!\frac{v}{2} – \sin^2\!\frac{v}{2}	+	\cos v = 1 – \sin^2\!\left(\frac{v}{2}\right) – \sin^2\!\left(\frac{v}{2}\right)
-	= 1 – 2\sin^2\!\frac{v}{2}</math>}}	+	= 1 – 2\sin^2\!\\left(frac{v}{2}\right)</math>}}
	so that		so that
	<div class="regel">		<div class="regel">
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
-	\sin^2\!\frac{v}{2} = \frac{1 – \cos v}{2}\,\mbox{.}</math>}}	+	\sin^2\!\left(\frac{v}{2}\right) = \frac{1 – \cos v}{2}\,\mbox{.}</math>}}
	</div>		</div>

-	Similarly, we can write <math>\sin^2(v/2)<\math> in terms of <math>\cos^2(v/2)</math>. Then we will arrive at	+	Similarly, we can write <math>\sin^2(v/2)</math> in terms of <math>\cos^2(v/2)</math>. Then we will arrive at

	<div class="regel">		<div class="regel">
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
-	\cos^2\!\frac{v}{2} = \frac{1 + \cos v}{2}\,\mbox{.}</math>}}	+	\cos^2\!\left(\frac{v}{2}\right) = \frac{1 + \cos v}{2}\,\mbox{.}</math>}}
	</div>		</div>

Jdinglis at 19:13, 10 November 2008

2008-11-10T19:13:58Z

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	== The addition, subtraction and double-angle formulas ==		== The addition, subtraction and double-angle formulas ==

-	~~One~~ often ~~needs~~ to deal with expressions in which two or more angles are involved, such as <math>\sin(u+v)</math>. ~~One~~ will then need the so-called "addition formulas" . For sine and cosine the formulas are	+	We often need to deal with expressions in which two or more angles are involved, such as <math>\sin(u+v)</math>. We will then need the so-called "addition formulas" . For sine and cosine the formulas are:

	<div class="regel">		<div class="regel">
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	</div>		</div>

-	If ~~one wants~~ to know the sine or cosine of a double angle, that is <math>\sin 2v</math> or <math>\cos 2v</math>, ~~one~~ can ~~write these expressions as <math>\sin(v + v)</math> or <math>\cos(v + v)</math> and~~ use the addition formulas above ~~and~~ get the double-angle formulas	+	If we want to know the sine or cosine of a double angle, that is <math>\sin 2v</math> or <math>\cos 2v</math>, we can use the addition formulas above to get the double-angle formulas:

	<div class="regel">		<div class="regel">
	{{Displayed math\|\|<math>\begin{align*}		{{Displayed math\|\|<math>\begin{align*}
	\sin 2v &= 2 \sin v \cos v\,\mbox{,}\\		\sin 2v &= 2 \sin v \cos v\,\mbox{,}\\
-	\cos 2v &= \cos^2\!v – \sin^2\!v \,\mbox{.}\\	+	\cos 2v &= \cos^2\!v\ –\ \sin^2\!v \,\mbox{.}\\
	\end{align*}</math>}}		\end{align*}</math>}}
	</div>		</div>

-	From these relationships, ~~one~~ can ~~then get the~~ formulas for half angles. By replacing <math>2v</math> by <math>v</math>, and consequently <math>v</math> by <math>v/2</math>, in the formula for <math>\cos 2v</math> ~~one gets~~ that	+	From these relationships, we can also derive formulas for half angles. By replacing <math>2v</math> by <math>v</math>, and consequently <math>v</math> by <math>v/2</math>, in the formula for <math>\cos 2v</math> we get that
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
-	\cos v = \cos^2\!\frac{v}{2} – \sin^2\!\frac{v}{2}\,\mbox{.}</math>}}	+	\cos v = \cos^2\!\frac{v}{2}\ –\ \sin^2\!\frac{v}{2}\,\mbox{.}</math>}}

-	If we want a formula for <math>\sin(v/2)</math> we use the Pythagorean identity to ~~get rid of~~ <math>\cos^2(v/2)</math>	+	If we want a formula for <math>\sin(v/2)</math> we use the Pythagorean identity to write <math>\cos^2(v/2)<\math> in terms of <math>\sin^2(v/2)</math>. Then
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\cos v = 1 – \sin^2\!\frac{v}{2} – \sin^2\!\frac{v}{2}		\cos v = 1 – \sin^2\!\frac{v}{2} – \sin^2\!\frac{v}{2}
	= 1 – 2\sin^2\!\frac{v}{2}</math>}}		= 1 – 2\sin^2\!\frac{v}{2}</math>}}
-	~~i.e.~~	+	so that
	<div class="regel">		<div class="regel">
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
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	</div>		</div>

-	Similarly, we can ~~use the Pythagorean identity to get rid of~~ <math>\sin^2(v/2)</math>. Then we will ~~have instead~~	+	Similarly, we can write <math>\sin^2(v/2)<\math> in terms of <math>\cos^2(v/2)</math>. Then we will arrive at

	<div class="regel">		<div class="regel">

Jdinglis at 19:04, 10 November 2008

2008-11-10T19:04:59Z

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	== Introduction ==		== Introduction ==

-	There is a variety of trigonometric formulas ~~to use if one wishes to transform between~~ the sine, cosine or tangent for an angle or multiples of an angle. They are usually known as the trigonometric identities~~, since they~~ ~~only lead to different ways to describe a single expression using a variety of trigonometric functions.~~ Here we will give some of these trigonometric relationships. There are many more than we can deal with in this course~~. Most~~ can be derived from the so-called Pythagorean identity and the addition and subtraction formulas ~~or identities~~ (see below), which are important to know by heart.	+	There are a variety of trigonometric formulas which relate the sine, cosine or tangent for an angle or multiples of an angle. They are usually known as the trigonometric identities. Here we will give some of these trigonometric relationships, and show how to derive them. There are many more than we can deal with in this course, but most can be derived from the so-called Pythagorean identity and the addition and subtraction formulas (see below), which are important to know by heart.


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	== Symmetries ==		== Symmetries ==

-	With the help of the unit circle ~~and reflection, and~~ exploiting the symmetries of the trigonometric functions one obtains a large ~~amount~~ of relationships between the cosine and sine functions.	+	With the help of the unit circle by exploiting the symmetries of the trigonometric functions one obtains a large number of relationships between the cosine and sine functions:

	<div class="regel">		<div class="regel">
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	</div>		</div>

-	Instead of trying to learn all of these relationships by heart, it ~~might~~ be better to learn how to derive them from the unit circle.	+	Instead of trying to learn all of these relationships by heart, it is be better to learn how to derive them from the unit circle.


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-	Reflection does not affect the ''x''- coordinate while the ''y''-coordinate changes sign	+	Reflection does not affect the ''x''- coordinate while the ''y''-coordinate changes sign, so that
	{{Displayed math\|\|<math>\begin{align*}		{{Displayed math\|\|<math>\begin{align*}
	\cos(-v) &= \cos v\,\mbox{,}\\		\cos(-v) &= \cos v\,\mbox{,}\\
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	<br>		<br>
-	Reflection in the ''y''-axis changes the angle <math>v</math> to <math>\pi-v</math> (the reflection makes an angle <math>v</math> with the negative ''x''-axis).	+	Reflection in the ''y''-axis changes the angle from <math>v</math> to <math>\pi-v</math> (the reflection makes an angle <math>v</math> with the negative ''x''-axis).


-	~~Reflection~~ does not affect the ''y''-coordinate while the ''x''-coordinate changes sign	+	This reflection does not affect the ''y''-coordinate, while the ''x''-coordinate changes sign, so we see that
	{{Displayed math\|\|<math>\begin{align*}		{{Displayed math\|\|<math>\begin{align*}
	\cos(\pi-v) &= -\cos v\,\mbox{,}\\		\cos(\pi-v) &= -\cos v\,\mbox{,}\\
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	<br>		<br>
-	The angle <math>v</math> ~~changes~~ to <math>\pi/2 - v</math> ( the ~~reflection~~ makes an angle <math>v</math> with the positive ''y''-axis).	+	The angle <math>v</math> is changed to <math>\pi/2 - v</math> (the reflected line makes an angle <math>v</math> with the positive ''y''-axis).


-	~~Reflection causes~~ the ''x''- and ''y''-coordinates ~~to change places~~	+	This reflection swaps the ''x''- and ''y''-coordinates, so that
	{{Displayed math\|\|<math>\begin{align*}		{{Displayed math\|\|<math>\begin{align*}
	\cos \Bigl(\frac{\pi}{2} - v \Bigr) &= \sin v\,\mbox{.}\\		\cos \Bigl(\frac{\pi}{2} - v \Bigr) &= \sin v\,\mbox{.}\\
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-	The rotation turns the ''x''-coordinate into the new ''y''-coordinate and the ''y''-~~coordinates~~ turns into the new ''x''-coordinate ~~though~~ with the opposite sign	+	The rotation turns the ''x''-coordinate into the new ''y''-coordinate and the ''y''-coordinate turns into the new ''x''-coordinate with the opposite sign, so that
	{{Displayed math\|\|<math>\begin{align*}		{{Displayed math\|\|<math>\begin{align*}
	\cos \Bigl(v+\frac{\pi}{2}\Bigr) &= -\sin v\,\mbox{,}\\		\cos \Bigl(v+\frac{\pi}{2}\Bigr) &= -\sin v\,\mbox{,}\\
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-	Alternatively, ~~one~~ can ~~get~~ these relationships by reflecting and / or ~~displacing graphs~~. For instance, if we want to ~~have a formula in which~~ <math>\cos v</math> ~~is expressed in terms of a~~ sine ~~one~~ can ~~displace~~ the graph ~~for cosine to fit~~ the ~~sine curve~~. This can be done in several ways, but the most natural is to write <math>\cos v = \sin (v + \pi / 2)</math>. To avoid mistakes, ~~one~~ can check that this is true for several different values of <math>v</math>.	+	Alternatively, we can derive these relationships by reflecting and/or shifting the graph of sine and cosine. For instance, if we want to express <math>\cos v</math> as the sine of an angle, we can shift the graph of <math>\sin v</math> along in the <math>v</math> direction, so that it coincides with the graph of <math>\cos v</math>. This can be done in several ways, but the most natural is to write <math>\cos v = \sin (v + \pi / 2)</math>. To avoid mistakes, we can check that this is true for several different values of <math>v</math>.

	<center>{{:4.3 - Figure - The curves y = cos x and y = sin x}}</center>		<center>{{:4.3 - Figure - The curves y = cos x and y = sin x}}</center>
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-	== The addition ~~and~~ subtraction ~~formulas~~ and double~~-angle and half~~-angle formulas ==	+	== The addition, subtraction and double-angle formulas ==

	One often needs to deal with expressions in which two or more angles are involved, such as <math>\sin(u+v)</math>. One will then need the so-called "addition formulas" . For sine and cosine the formulas are		One often needs to deal with expressions in which two or more angles are involved, such as <math>\sin(u+v)</math>. One will then need the so-called "addition formulas" . For sine and cosine the formulas are

Tekbot: Robot: Automated text replacement (-{{:4.3 - Figur - Vridning med vinkeln π/2}} +{{:4.3 - Figure - Rotation by an angle π/2}})

2008-09-12T13:42:12Z

Robot: Automated text replacement (-{{:4.3 - Figur - Vridning med vinkeln π/2}} +{{:4.3 - Figure - Rotation by an angle π/2}})

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-	{{:4.3 - ~~Figur~~ - ~~Vridning med vinkeln~~ π/2}}	+	{{:4.3 - Figure - Rotation by an angle π/2}}
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Tekbot: Robot: Automated text replacement (-{{:4.3 - Figur - Trigonometriska ettan}} +{{:4.3 - Figure - Pythagorean identity}})

2008-09-12T13:41:10Z

Robot: Automated text replacement (-{{:4.3 - Figur - Trigonometriska ettan}} +{{:4.3 - Figure - Pythagorean identity}})

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	which is usually written as <math>\sin^2\!v + \cos^2\!v = 1</math>.		which is usually written as <math>\sin^2\!v + \cos^2\!v = 1</math>.
	\| valign="center" \|		\| valign="center" \|
-	{{:4.3 - ~~Figur~~ - ~~Trigonometriska ettan~~}}	+	{{:4.3 - Figure - Pythagorean identity}}
	\|}		\|}