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<h1>MyPhysicsLab – Math Refresher</h1>
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Here is a quick math refresher on calculus and trig, to help you enjoy the math behind the <a href="http://www.myphysicslab.com/index.html">physics simulations</a> at MyPhysicsLab.
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<h2 id="derivatives">Derivatives</h2>
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The notation for the first derivative of a function <span class="inline_eqn"><i>x</i>(<i>t</i>)</span>, with respect to the variable <span class="inline_eqn"><i>t</i></span>, can be written as
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<span class="display_eqn" style="line-height: normal;">
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<i>x'</i><br>
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<i>x'</i>(<i>t</i>)<br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <span class="inline_eqn"><i>x</i>(<i>t</i>)</span><br>
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</span>
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These are all equivalent. The notation for the second derivative is <span class="inline_eqn"><i>x''</i></span> or <span class="inline_eqn"><i>x''</i>(<i>t</i>)</span>.<br>
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<br>
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Here are some of the basic rules for calculating derivatives. In the following <i>k</i> and <i>n</i> are real non-zero constants. And <span class="inline_eqn"><i>h</i>(<i>t</i>), <i>g</i>(<i>t</i>)</span> are functions of <span class="inline_eqn"><i>t</i></span>.<br>
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<br>
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First consider powers of <i>t</i>. The general rule is
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<span class="display_eqn">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>t</i> <sup><i>n</i></sup> = <i>n</i> <i>t</i> <sup><i>n</i> − 1</sup> <span class="text"> for any <i>n</i> ≠ 0</span><br>
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</span>
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Here are some examples of derivatives of powers, using the above rule:
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<span class="display_eqn">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>t</i> = 1<br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>t</i> <sup>2</sup> = 2 <i>t</i><br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> (<i>t</i> <sup>3</sup> + <i>t</i> <sup>2</sup> + <i>t</i> + 1) = 3t <sup>2</sup> + 2t + 1<br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <sup>1</sup>⁄<sub><i>t</i></sub> = <sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>t</i> <sup>−1</sup> = − <i>t</i> <sup>−2</sup> = −<sup>1</sup>⁄<sub><i>t</i> <sup>2</sup></sub><br>
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</span>
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Here are some basic rules about derivatives:
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<span class="display_eqn">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>k</i> = 0 <span class="text"> (<i>k</i> = constant)</span><br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <span class="bigg">(</span><i>k</i> <i>h</i>(<i>t</i>)<span class="bigg">)</span> = <i>k</i> <sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>h</i>(<i>t</i>) <span class="text"> (<i>k</i> = constant)</span><br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <span class="bigg">(</span><i>h</i>(<i>t</i>) + <i>g</i>(<i>t</i>)<span class="bigg">)</span> = <sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>h</i>(<i>t</i>) +
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>g</i>(<i>t</i>)
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<br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <span class="bigg">(</span><i>h</i>(<i>t</i>) <20> <i>g</i>(<i>t</i>)<span class="bigg">)</span> = <i>h</i><EFBFBD><i>g'</i> + <i>h'</i><EFBFBD><i>g</i> <span class="text"> The product rule</span><br>
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</span>
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Here are derivatives of some very important special functions
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<span class="display_eqn">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> sin(<i>t</i>) = cos(<i>t</i>)<br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> cos(<i>t</i>) = −sin(<i>t</i>)<br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>e</i> <sup><i>t</i></sup> = <i>e</i> <sup><i>t</i></sup><br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> ln(<i>t</i>) = <sup>1</sup>⁄<sub><i>t</i></sub> <span class="text"> Natural logarithm</span><br>
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</span>
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The all-important <em>chain rule</em> lets us take the derivative of <em>functions of functions</em> (also called <em>function composition</em>):
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<span class="display_eqn">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>h</i>(<i>g</i>(<i>t</i>)) = <i>h'</i>(<i>g</i>(<i>t</i>)) <20> <i>g'</i>(<i>t</i>)
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<span class="text"> The chain rule</span><br>
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</span>
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It's important to get good at using the chain rule. Here are some examples of the chain rule in action:
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<span class="display_eqn">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> sin(<i>h</i>(<i>t</i>)) = cos(<i>h</i>(<i>t</i>)) <i>h'</i>(<i>t</i>)<br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> sin(<i>t</i> <sup>2</sup>) = 2 <i>t</i> cos(<i>t</i> <sup>2</sup>)<br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>e</i> <sup><i>h</i>(<i>t</i>)</sup> = <i>h'</i>(<i>t</i>) <i>e</i> <sup><i>h</i>(<i>t</i>)</sup><br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>e</i> <sup><i>k</i> <i>t</i></sup> = <i>k</i> <i>e</i> <sup><i>k</i> <i>t</i></sup> <span class="text"> (<i>k</i> = constant)</span><br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <i>e</i> <sup><i>t</i> <sup>2</sup></sup> = 2 <i>t</i> <i>e</i> <sup><i>t</i> <sup>2</sup></sup><br>
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> ln(<i>h</i>(<i>t</i>)) = <i>h'</i>(<i>t</i>) ⁄ <i>h</i>(<i>t</i>)<br>
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</span>
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<table class="fraction" align="center" cellpadding="0" cellspacing="0">
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<tbody><tr>
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<td rowspan="2">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <span class="bigg">(</span>
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</td><td nowrap="nowrap">
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1
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</td><td rowspan="2">
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<span class="bigg">)</span> =
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</td><td nowrap="nowrap">
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− <i>h'</i>(<i>t</i>)
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</td>
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</tr><tr>
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<td class="upper_line" nowrap="nowrap">
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<i>h</i>(<i>t</i>)
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</td><td class="upper_line" nowrap="nowrap">
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<i>h</i>(t)<sup>2</sup>
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</td>
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</tr>
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</tbody></table>
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The quotient rule gives us the derivative of a <em>ratio of functions</em>:
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<table class="fraction" align="center" cellpadding="0" cellspacing="0">
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<tbody><tr>
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<td rowspan="2">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <span class="bigg">(</span>
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</td><td nowrap="nowrap">
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<i>h</i>(<i>t</i>)
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</td><td rowspan="2">
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<span class="bigg">)</span> =
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</td><td nowrap="nowrap">
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<i>g</i> <i>h'</i> − <i>h</i> <i>g'</i>
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</td><td rowspan="2">
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<span class="text"> The quotient rule</span>
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</td>
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</tr><tr>
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<td class="upper_line" nowrap="nowrap">
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<i>g</i>(<i>t</i>)
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</td><td class="upper_line" nowrap="nowrap">
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<i>g</i> <sup>2</sup>
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</td>
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</tr>
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</tbody></table>
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Using the chain rule and the product rule we can derive the quotient rule:
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<span class="display_eqn">
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<sup><i>d</i></sup>⁄<sub><i>dt</i></sub> <sup><i>h</i>(<i>t</i>)</sup>⁄<sub><i>g</i>(<i>t</i>)</sub> =
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<span class="nobr"><sup><i>d</i></sup>⁄<sub><i>dt</i></sub> (<i>h</i> <20> <sup>1</sup>⁄<sub><i>g</i></sub>)</span> =
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<span class="nobr"><i>h'</i> <20> (<sup>1</sup>⁄<sub><i>g</i></sub>)</span> +
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<span class="nobr"><i>h</i> <20> (<sup>−<i>g'</i></sup>⁄<sub><i>g</i><sup>2</sup></sub>)</span> =
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<span class="nobr">(<i>g</i> <i>h'</i> − <i>h</i> <i>g'</i>) ⁄ <i>g</i><sup>2</sup></span>
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</span>
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<h2 id="trig_identities">Trig Identities</h2>
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First, a note on some confusing notation: an exponent of <span class="inline_eqn">−1</span> on a trig function means the <i>inverse</i> of that function (not the reciprocal!). Therefore
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<span class="display_eqn">
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tan<sup>−1</sup>(<i>x</i>) = arctan(<i>x</i>)
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</span>
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while
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<span class="display_eqn">
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tan<sup>2</sup>(<i>x</i>) = (tan(<i>x</i>))<sup>2</sup>
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</span>
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The best way to get comfortable with trigonometry is to think in terms
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of the unit circle. Most of these identities then become obvious.
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<span class="display_eqn">
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sin(−<i>x</i>) = −sin <i>x</i><br>
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cos(−<i>x</i>) = cos <i>x</i><br>
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tan(−<i>x</i>) = −tan <i>x</i><br>
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sin <i>x</i> = cos(<sup>π</sup>⁄<sub>2</sub> − <i>x</i>)<br>
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cos <i>x</i> = sin(<sup>π</sup>⁄<sub>2</sub> − <i>x</i>)<br>
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sin(0) = 0<br>
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cos(0) = 1<br>
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sin(<sup>π</sup>⁄<sub>2</sub>) = 1<br>
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cos(<sup>π</sup>⁄<sub>2</sub>) = 0<br>
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sin(π) = 0<br>
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cos(π) = −1<br>
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sin(<sup>3π</sup>⁄<sub>2</sub>) = −1<br>
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cos(<sup>3π</sup>⁄<sub>2</sub>) = 0<br>
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sin(<i>x</i> + 2<i>n</i>π) = sin <i>x</i> <span class="text"> <i>n</i> an integer</span><br>
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cos(<i>x</i> + 2<i>n</i>π) = cos <i>x</i> <span class="text"> <i>n</i> an integer</span><br>
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</span>
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The famous <em>pythagorean theorem</em> gives us the following identity
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<span class="display_eqn">
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cos<sup>2</sup><i>x</i> + sin<sup>2</sup><i>x</i> = 1<br>
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</span>
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The <em><a id="sum_of_angles">sum of angles</a></em> formulas are
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<span class="display_eqn">
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cos(<i>x</i> + <i>y</i>) = cos <i>x</i> cos <i>y</i> − sin <i>x</i> sin <i>y</i><br>
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cos(<i>x</i> − <i>y</i>) = cos <i>x</i> cos <i>y</i> + sin <i>x</i> sin <i>y</i><br>
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sin(<i>x</i> + <i>y</i>) = sin <i>x</i> cos <i>y</i> + cos <i>x</i> sin <i>y</i><br>
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sin(<i>x</i> − <i>y</i>) = sin <i>x</i> cos <i>y</i> − cos <i>x</i> sin <i>y</i><br>
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</span>
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<20> <a href="mailto:erikn@MyPhysicsLab.com" title="send comments to Erik Neumann">Erik Neumann</a>, 2004
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