Aus Online Mathematik Brückenkurs 1

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__NOTOC__ {| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | style="border-bottom:1px solid #797979" width="5px" | &nbsp; {{Gewählter Tab|[[4.3 Trigonometric relations|Theorie]]}} {{Nicht gewählter Tab|[[4.3 Übungen|Übungen]]}} | style="border-bottom:1px solid #797979" width="100%"| &nbsp; |} {{Info| '''Inhalt:''' * The Pythagorean identity * The double-angle and half-angle formulas * Addition and subtraction formulas }} {{Info| '''Lernziele:''' Nach diesem Abschnitt sollst Du folgendes können how to: *Derive trigonometric relationships from symmetries in the unit circle. * Simplify trigonometric expressions with the help of trigonometric formulas. }} == 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. == The Pythagorean identity == {| width="100%" | width="100%" valign="center" | This identity is the most basic, but is in fact nothing more than Pythagorean theorem, applied to the unit circle. The right-angled triangle on the right shows that {{Abgesetzte Formel||<math>(\sin v)^2 + (\cos v)^2 = 1\,\mbox{,}</math>}} which is usually written as <math>\sin^2\!v + \cos^2\!v = 1</math>. | valign="center" | {{:4.3 - Bild - Sats des Pythagoras}} |} == 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. <div class="regel"> {{Abgesetzte Formel||<math> \begin{align*} \cos (-v) &= \cos v\vphantom{\Bigl(}\\ \sin (-v) &= - \sin v\vphantom{\Bigl(}\\ \cos (\pi-v) &= - \cos v\vphantom{\Bigl(}\\ \sin (\pi-v) &= \sin v\vphantom{\Bigl(}\\ \end{align*} \qquad\quad \begin{align*} \cos \Bigl(\displaystyle \frac{\pi}{2} -v \Bigr) &= \sin v\\ \sin \Bigl(\displaystyle \frac{\pi}{2} -v \Bigr) &= \cos v\\ \cos \Bigl(v + \displaystyle \frac{\pi}{2} \Bigr) &= - \sin v\\ \sin \Bigl( v + \displaystyle \frac{\pi}{2} \Bigr) &= \cos v\\ \end{align*} </math>}} </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. '''Reflection in the ''x''-axis''' {| |- | width=50% valign=top align="center" | {{:4.3 - Bild - Reflektion in der x-Achse}} | width=45% valign=top | <br> When an angle <math>v</math> is reflected in the ''x''-axis it becomes<math>-v</math>. Reflection does not affect the ''x''- coordinate while the ''y''-coordinate changes sign {{Abgesetzte Formel||<math>\begin{align*} \cos(-v) &= \cos v\,\mbox{,}\\ \sin (-v) &= - \sin v\,\mbox{.}\\ \end{align*}</math>}} | width=5% valign=top | |} '''Reflection in the ''y''-axis''' {| |- | width=50% valign=top align="center" | {{:4.3 - Bild - Reflektion in der y-Achse}} | width=45% valign=top | <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 does not affect the ''y''-coordinate while the ''x''-coordinate changes sign {{Abgesetzte Formel||<math>\begin{align*} \cos(\pi-v) &= -\cos v\,\mbox{,}\\ \sin (\pi-v) &= \sin v\,\mbox{.}\\ \end{align*}</math>}} | width=5% valign=top | |} ''' Reflection in the line ''y = x'' ''' {| |- | width=50% valign=top align="center" | {{:4.3 - Bild - Reflektion in der Geraden y = x}} | width=45% valign=top | <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). Reflection causes the ''x''- and ''y''-coordinates to change places {{Abgesetzte Formel||<math>\begin{align*} \cos \Bigl(\frac{\pi}{2} - v \Bigr) &= \sin v\,\mbox{.}\\ \sin \Bigl(\frac{\pi}{2} - v \Bigr) &= \cos v\,\mbox{.}\\ \end{align*}</math>}} | width=5% valign=top | |} ''' Rotation by an angle of <math>\mathbf{\pi/2}</math>''' {| |- | width=50% valign=top align="center" | {{:4.3 - Bild - Rotation mit dem Winkel π/2}} | width=40% valign=top | <br> A rotation <math>\pi/2</math> of the angle <math>v</math> means that the angle becomes <math>v+\pi/2</math>. 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 {{Abgesetzte Formel||<math>\begin{align*} \cos \Bigl(v+\frac{\pi}{2}\Bigr) &= -\sin v\,\mbox{,}\\ \sin \Bigl(v+\frac{\pi}{2} \Bigr) &= \cos v\,\mbox{.} \end{align*}</math>}} | width=10% valign=top | |} 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>. <center>{{:4.3 - Bild - Die Kurven y = cos x und y = sin x}}</center> Check: <math>\ \cos 0 = \sin (0 + \pi / 2)=1</math>. == The addition and subtraction formulas and double-angle and half-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 <div class="regel"> {{Abgesetzte Formel||<math>\begin{align*} \sin(u + v) &= \sin u\,\cos v + \cos u\,\sin v\,\mbox{,}\\ \sin(u – v) &= \sin u\,\cos v – \cos u\,\sin v\,\mbox{,}\\ \cos(u + v) &= \cos u\,\cos v – \sin u\,\sin v\,\mbox{,}\\ \cos(u – v) &= \cos u\,\cos v + \sin u\,\sin v\,\mbox{.}\\ \end{align*}</math>}} </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 <div class="regel"> {{Abgesetzte Formel||<math>\begin{align*} \sin 2v &= 2 \sin v \cos v\,\mbox{,}\\ \cos 2v &= \cos^2\!v – \sin^2\!v \,\mbox{.}\\ \end{align*}</math>}} </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 {{Abgesetzte Formel||<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> {{Abgesetzte Formel||<math> \cos v = 1 – \sin^2\!\frac{v}{2} – \sin^2\!\frac{v}{2} = 1 – 2\sin^2\!\frac{v}{2}</math>}} i.e. <div class="regel"> {{Abgesetzte Formel||<math> \sin^2\!\frac{v}{2} = \frac{1 – \cos v}{2}\,\mbox{.}</math>}} </div> Similarly, we can use the Pythagorean identity to get rid of <math>\sin^2(v/2)</math>. Then we will have instead <div class="regel"> {{Abgesetzte Formel||<math> \cos^2\!\frac{v}{2} = \frac{1 + \cos v}{2}\,\mbox{.}</math>}} </div> [[4.3 Übungen|Übungen]] <div class="inforuta" style="width:580px;"> '''Tipps fürs lernen''' '''The basic and final tests''' Nachdem Du fertig mit der Theorie bist, sollst Du die diagnostische Prüfung und die Schlussprüfung machen. Du findest die links zu den Prüfungen in Deiner "Student Lounge". '''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 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. '''Useful web sites''' [http://www.ies.co.jp/math/java/trig/cosbox/cosbox.html Experiment with the cosine “box” ] </div>

• 4.3 - Bild - Reflektion in der Geraden y = x
• 4.3 - Bild - Reflektion in der x-Achse
• 4.3 - Bild - Reflektion in der y-Achse
• 4.3 - Bild - Rotation mit dem Winkel π/2
• 4.3 - Bild - Sats des Pythagoras
• Vorlage:Abgesetzte Formel
• Vorlage:Gewählter Tab
• Vorlage:Info
• Vorlage:Nicht gewählter Tab

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