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<!DOCTYPE html>
<html data-require="math graphie convert-values">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Pythagorean identities</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="sincos">
<div class="vars">
<var id="OPTIONS">
shuffle(["\\sin\\theta", "\\cos\\theta",
"\\tan\\theta", "\\sec\\theta", "\\csc\\theta",
"\\cot\\theta"]).slice(0,3)
</var>
<var id="FUNC">randFromArray(OPTIONS)</var>
<var id="MULT">random() &lt; 0.5</var>
</div>
<p class="question">
</p><div data-if="MULT">
<code>(\sin^2 \theta + \cos^2 \theta)(<var>FUNC</var>)
= \; ?</code>
</div>
<div data-else="">
<code>\dfrac{<var>FUNC</var>}
{\sin^2 \theta + \cos^2 \theta} = \; ?</code>
</div>
<p></p>
<div class="solution"><code><var>FUNC</var></code></div>
<ul class="choices" data-show="3">
<li data-each="OPTIONS as op"><code><var>op</var></code></li>
</ul>
<div class="hints">
<div>
<p>We can use the identity
<code>\blue{\sin^2 \theta} + \orange{\cos^2 \theta}
= 1</code>
to simplify this expression.</p>
<div class="graphie">
init({
range: [[-1.2, 1.2], [-1.3, 1.3]],
scale: 130
});
with(KhanUtil.currentGraph) {
style({
stroke: "#ddd",
strokeWidth: 1,
arrows: "-&gt;"
});
circle([0, 0], 1);
line([-1.2, 0], [1.2, 0]);
line([0, -1.2], [0, 1.2]);
line([1.2, 0], [-1.2, 0]);
line([0, 1.2], [0, -1.2]);
style({
strokeWidth: 2.5,
arrows: ""
});
ang = 2*Math.PI/3;
line([0, 0], [cos(ang), sin(ang)],
{stroke: "black"});
label([cos(ang)/2, sin(ang)/2],
"1", "above right");
line([0, 0], [cos(ang), 0],
{stroke: ORANGE});
label([cos(ang), sin(ang)/2],
"\\blue{\\sin\\theta}", "left");
line([cos(ang), 0],
[cos(ang), sin(ang)],
{stroke: BLUE});
label([cos(ang)/2, 0],
"\\orange{\\cos\\theta}", "below");
arc([0,0], 0.2, 0, 120,
{stroke: "black", arrows: "-&gt;"});
label([0,0.1], "\\theta", "above right");
}
</div>
<p>We can see why this is true by using the
Pythagorean Theorem.</p>
</div>
<p data-if="MULT">
So, <code>(\sin^2 \theta + \cos^2 \theta)(<var>FUNC</var>)
= 1 \cdot <var>FUNC</var> = <var>FUNC</var></code>
</p>
<p data-else="">
So, <code>\dfrac{<var>FUNC</var>}
{\sin^2 \theta + \cos^2 \theta} =
\dfrac{<var>FUNC</var>}{1} = <var>FUNC</var></code>
</p>
</div>
</div>
<div id="1MinusSinCos">
<div class="vars">
<var id="MULT">random() &lt; 0.5</var>
<var id="SIN">random() &lt; 0.5</var>
<var id="IDENT, EQUIV">
SIN ? ["1 - \\sin^2\\theta", "\\cos^2\\theta"]
: ["1 - \\cos^2\\theta", "\\sin^2\\theta"]
</var>
<var id="OPTIONS, FUNC, ANS">
trig.getOptionsResult(EQUIV, (MULT ? "*" : "/"))
</var>
<var id="FUNC_SIMP">
MULT ?
trig.showSimplified(FUNC) : trig.showSimplified(FUNC, true)
</var>
</div>
<p class="question">
</p><div data-if="MULT">
<code>(<var>IDENT</var>)(<var>FUNC</var>) = \; ?</code>
</div>
<div data-else="">
<code>\dfrac{<var>IDENT</var>}{<var>FUNC</var>}
= \; ?</code>
</div>
<p></p>
<div class="solution"><code><var>ANS</var></code>
</div>
<ul class="choices" data-show="3">
<li data-each="OPTIONS as op"><code><var>op</var></code></li>
</ul>
<div class="hints">
<div>
<p>We can use the identity
<code>\blue{\sin^2 \theta} + \orange{\cos^2 \theta}
= 1</code>
to simplify this expression.</p>
<div class="graphie">
init({
range: [[-1.2, 1.2], [-1.3, 1.3]],
scale: 130
});
with(KhanUtil.currentGraph) {
style({
stroke: "#ddd",
strokeWidth: 1,
arrows: "-&gt;"
});
circle([0, 0], 1);
line([-1.2, 0], [1.2, 0]);
line([0, -1.2], [0, 1.2]);
line([1.2, 0], [-1.2, 0]);
line([0, 1.2], [0, -1.2]);
style({
strokeWidth: 2.5,
arrows: ""
});
ang = 2*Math.PI/3;
line([0, 0], [cos(ang), sin(ang)],
{stroke: "black"});
label([cos(ang)/2, sin(ang)/2],
"1", "above right");
line([0, 0], [cos(ang), 0],
{stroke: ORANGE});
label([cos(ang), sin(ang)/2],
"\\blue{\\sin\\theta}", "left");
line([cos(ang), 0],
[cos(ang), sin(ang)],
{stroke: BLUE});
label([cos(ang)/2, 0],
"\\orange{\\cos\\theta}", "below");
arc([0,0], 0.2, 0, 120,
{stroke: "black", arrows: "-&gt;"});
label([0,0.1], "\\theta", "above right");
}
</div>
<p>We can see why this is true by using the
Pythagorean Theorem.</p>
</div>
<div>
<p>So, <code><var>IDENT</var> = <var>EQUIV</var></code></p>
</div>
<div>
<p>Plugging into our expression, we get</p>
<div data-if="MULT">
<p><code>\qquad
(<var>IDENT</var>)(<var>FUNC</var>)
=
(<var>EQUIV</var>)(<var>FUNC</var>)
</code></p>
</div>
<div data-else="">
<p><code>\qquad
\dfrac{<var>IDENT</var>}{<var>FUNC</var>}
=
\dfrac{<var>EQUIV</var>}{<var>FUNC</var>}
</code></p>
</div>
</div>
<div data-if="FUNC !== '\\sin^2\\theta' && FUNC !== '\\cos^2\\theta'">
<p>To make simplifying easier, let's put everything
in terms of <code>\sin</code> and <code>\cos</code>.
<code><var>FUNC</var> = <var>FUNC_SIMP</var></code>,
so we can plug that in to get</p>
<div data-if="MULT">
<p><code>\qquad
(<var>EQUIV</var>)(<var>FUNC</var>)
=
\left(<var>EQUIV</var>\right)
\left(<var>FUNC_SIMP</var>\right)
</code></p>
</div>
<div data-else="">
<p><code>\qquad
\dfrac{<var>EQUIV</var>}{<var>FUNC</var>}
=
\dfrac{<var>EQUIV</var>}{<var>FUNC_SIMP</var>}
</code></p>
</div>
</div>
<div>
This is <code><var>ANS</var></code>.
</div>
</div>
</div>
<div id="1PlusTan">
<div class="vars">
<var id="MULT">random() &lt; 0.5</var>
<var id="TAN">random() &lt; 0.5</var>
<var id="IDENT, EQUIV">
TAN ? ["\\tan^2\\theta + 1", "\\sec^2\\theta"]
: ["\\sec^2\\theta-1", "\\tan^2\\theta"]
</var>
<var id="OPTIONS, FUNC, ANS">
trig.getOptionsResult(EQUIV, (MULT ? "*" : "/"))
</var>
<var id="FUNC_SIMP, EQUIV_SIMP">
[trig.showSimplified(FUNC, !MULT),
trig.showSimplified(EQUIV, !MULT)]
</var>
<var id="ANS_SIMP">
trig.showSimplified(ANS)
</var>
</div>
<p class="question">
</p><div data-if="MULT">
<code>(<var>IDENT</var>)(<var>FUNC</var>) = \; ?</code>
</div>
<div data-else="">
<code>\dfrac{<var>IDENT</var>}{<var>FUNC</var>}
= \; ?</code>
</div>
<p></p>
<div class="solution"><code><var>ANS</var></code>
</div>
<ul class="choices" data-show="3">
<li data-each="OPTIONS as op"><code><var>op</var></code></li>
</ul>
<div class="hints">
<div>
<p>We can derive a useful identity from
<code>\blue{\sin^2 \theta} + \orange{\cos^2 \theta}
= 1</code>
to simplify this expression.</p>
<div class="graphie">
init({
range: [[-1.2, 1.2], [-1.3, 1.3]],
scale: 130
});
with(KhanUtil.currentGraph) {
style({
stroke: "#ddd",
strokeWidth: 1,
arrows: "-&gt;"
});
circle([0, 0], 1);
line([-1.2, 0], [1.2, 0]);
line([0, -1.2], [0, 1.2]);
line([1.2, 0], [-1.2, 0]);
line([0, 1.2], [0, -1.2]);
style({
strokeWidth: 2.5,
arrows: ""
});
ang = 2*Math.PI/3;
line([0, 0], [cos(ang), sin(ang)],
{stroke: "black"});
label([cos(ang)/2, sin(ang)/2],
"1", "above right");
line([0, 0], [cos(ang), 0],
{stroke: ORANGE});
label([cos(ang), sin(ang)/2],
"\\blue{\\sin\\theta}", "left");
line([cos(ang), 0],
[cos(ang), sin(ang)],
{stroke: BLUE});
label([cos(ang)/2, 0],
"\\orange{\\cos\\theta}", "below");
arc([0,0], 0.2, 0, 120,
{stroke: "black", arrows: "-&gt;"});
label([0,0.1], "\\theta", "above right");
}
</div>
<p>We can see why this identity is true by using the
Pythagorean Theorem.</p>
</div>
<div>
<p>Dividing both sides by <code>\cos^2\theta</code>, we get</p>
<p><code>\qquad \dfrac{\sin^2\theta}{\cos^2\theta}
+ \dfrac{\cos^2\theta}{\cos^2\theta}
= \dfrac{1}{\cos^2\theta}</code></p>
<div data-if="!TAN">
<code>\qquad \tan^2\theta + 1 = \sec^2\theta</code>
</div>
<p><code>\qquad <var>IDENT</var>
= <var>EQUIV</var></code></p>
</div>
<div>
<p>Plugging into our expression, we get</p>
<div data-if="MULT">
<p><code>\qquad
(<var>IDENT</var>)(<var>FUNC</var>)
=
\left(<var>EQUIV</var>\right)
\left(<var>FUNC</var>\right)
</code></p>
</div>
<div data-else="">
<p><code>\qquad
\dfrac{<var>IDENT</var>}{<var>FUNC</var>}
=
\dfrac{<var>EQUIV</var>}{<var>FUNC</var>}
</code></p>
</div>
</div>
<div data-if="FUNC !== '\\sin^2\\theta' && FUNC !== '\\cos^2\\theta'">
<p>To make simplifying easier, let's put everything
in terms of <code>\sin</code> and <code>\cos</code>.
We know <code><var>EQUIV</var>
= <var>EQUIV_SIMP</var></code>
and <code><var>FUNC</var> = <var>FUNC_SIMP</var></code>,
so we can substitute to get</p>
<div data-if="MULT">
<p><code>\qquad
\left(<var>EQUIV</var>\right)
\left(<var>FUNC</var>\right)
=
\left(<var>EQUIV_SIMP</var>\right)
\left(<var>FUNC_SIMP</var>\right)
</code></p>
</div>
<div data-else="">
<p><code>\qquad
\dfrac{<var>EQUIV</var>}{<var>FUNC</var>}
=
\dfrac{<var>EQUIV_SIMP</var>}{<var>FUNC_SIMP</var>}
</code></p>
</div>
</div>
<div data-else="">
<p>To make simplifying easier, let's put everything
in terms of <code>\sin</code> and <code>\cos</code>.
We know <code><var>EQUIV</var>
= <var>EQUIV_SIMP</var></code>, so we can substitute
to get</p>
<div data-if="MULT">
<p><code>\qquad
\left(<var>EQUIV</var>\right)
\left(<var>FUNC</var>\right)
=
\left(<var>EQUIV_SIMP</var>\right)
\left(<var>FUNC_SIMP</var>\right)
</code></p>
</div>
<div data-else="">
<p><code>\qquad
\dfrac{<var>EQUIV</var>}{<var>FUNC</var>}
=
\dfrac{<var>EQUIV_SIMP</var>}{<var>FUNC_SIMP</var>}
</code></p>
</div>
</div>
<div data-if="ANS !== '\\sin^2\\theta' && ANS !== '\\cos^2\\theta' && ANS !== '1'">
This is <code><var>ANS_SIMP</var> = <var>ANS</var></code>.
</div>
<div data-else="">
This is <code><var>ANS</var></code>.
</div>
</div>
</div>
<div id="1PlusCot">
<div class="vars">
<var id="MULT">random() &lt; 0.5</var>
<var id="COT">random() &lt; 0.5</var>
<var id="IDENT, EQUIV">
COT ? ["1 + \\cot^2\\theta", "\\csc^2\\theta"]
: ["\\csc^2\\theta-1", "\\cot^2\\theta"]
</var>
<var id="OPTIONS, FUNC, ANS">
trig.getOptionsResult(EQUIV, (MULT ? "*" : "/"))
</var>
<var id="FUNC_SIMP, EQUIV_SIMP">
[trig.showSimplified(FUNC, !MULT),
trig.showSimplified(EQUIV, !MULT)]
</var>
<var id="ANS_SIMP">
trig.showSimplified(ANS)
</var>
</div>
<p class="question">
</p><div data-if="MULT">
<code>(<var>IDENT</var>)(<var>FUNC</var>) = \; ?</code>
</div>
<div data-else="">
<code>\dfrac{<var>IDENT</var>}{<var>FUNC</var>}
= \; ?</code>
</div>
<p></p>
<div class="solution"><code><var>ANS</var></code>
</div>
<ul class="choices" data-show="3">
<li data-each="OPTIONS as op"><code><var>op</var></code></li>
</ul>
<div class="hints">
<div>
<p>We can derive a useful identity from
<code>\blue{\sin^2 \theta} + \orange{\cos^2 \theta}
= 1</code>
to simplify this expression.</p>
<div class="graphie">
init({
range: [[-1.2, 1.2], [-1.3, 1.3]],
scale: 130
});
with(KhanUtil.currentGraph) {
style({
stroke: "#ddd",
strokeWidth: 1,
arrows: "-&gt;"
});
circle([0, 0], 1);
line([-1.2, 0], [1.2, 0]);
line([0, -1.2], [0, 1.2]);
line([1.2, 0], [-1.2, 0]);
line([0, 1.2], [0, -1.2]);
style({
strokeWidth: 2.5,
arrows: ""
});
ang = 2*Math.PI/3;
line([0, 0], [cos(ang), sin(ang)],
{stroke: "black"});
label([cos(ang)/2, sin(ang)/2],
"1", "above right");
line([0, 0], [cos(ang), 0],
{stroke: ORANGE});
label([cos(ang), sin(ang)/2],
"\\blue{\\sin\\theta}", "left");
line([cos(ang), 0],
[cos(ang), sin(ang)],
{stroke: BLUE});
label([cos(ang)/2, 0],
"\\orange{\\cos\\theta}", "below");
arc([0,0], 0.2, 0, 120,
{stroke: "black", arrows: "-&gt;"});
label([0,0.1], "\\theta", "above right");
}
</div>
<p>We can see why this identity is true by using the
Pythagorean Theorem.</p>
</div>
<div>
<p>Dividing both sides by <code>\sin^2\theta</code>, we get</p>
<p><code>\qquad \dfrac{\sin^2\theta}{\sin^2\theta}
+ \dfrac{\cos^2\theta}{\sin^2\theta}
= \dfrac{1}{\sin^2\theta}</code></p>
<div data-if="!COT">
<code>\qquad 1 + \cot^2\theta = \csc^2\theta</code>
</div>
<p><code>\qquad <var>IDENT</var>
= <var>EQUIV</var></code></p>
</div>
<div>
<p>Plugging into our expression, we get</p>
<div data-if="MULT">
<p><code>\qquad
(<var>IDENT</var>)(<var>FUNC</var>)
=
\left(<var>EQUIV</var>\right)
\left(<var>FUNC</var>\right)
</code></p>
</div>
<div data-else="">
<p><code>\qquad
\dfrac{<var>IDENT</var>}{<var>FUNC</var>}
=
\dfrac{<var>EQUIV</var>}{<var>FUNC</var>}
</code></p>
</div>
</div>
<div data-if="FUNC !== '\\sin^2\\theta' && FUNC !== '\\cos^2\\theta'">
<p>To make simplifying easier, let's put everything
in terms of <code>\sin</code> and <code>\cos</code>.
We know <code><var>EQUIV</var>
= <var>EQUIV_SIMP</var></code>
and <code><var>FUNC</var> = <var>FUNC_SIMP</var></code>,
so we can substitute to get</p>
<div data-if="MULT">
<p><code>\qquad
\left(<var>EQUIV</var>\right)
\left(<var>FUNC</var>\right)
=
\left(<var>EQUIV_SIMP</var>\right)
\left(<var>FUNC_SIMP</var>\right)
</code></p>
</div>
<div data-else="">
<p><code>\qquad
\dfrac{<var>EQUIV</var>}{<var>FUNC</var>}
=
\dfrac{<var>EQUIV_SIMP</var>}{<var>FUNC_SIMP</var>}
</code></p>
</div>
</div>
<div data-else="">
<p>To make simplifying easier, let's put everything
in terms of <code>\sin</code> and <code>\cos</code>.
We know <code><var>EQUIV</var>
= <var>EQUIV_SIMP</var></code>, so we can substitute
to get</p>
<div data-if="MULT">
<p><code>\qquad
\left(<var>EQUIV</var>\right)
\left(<var>FUNC</var>\right)
=
\left(<var>EQUIV_SIMP</var>\right)
\left(<var>FUNC_SIMP</var>\right)
</code></p>
</div>
<div data-else="">
<p><code>\qquad
\dfrac{<var>EQUIV</var>}{<var>FUNC</var>}
=
\dfrac{<var>EQUIV_SIMP</var>}{<var>FUNC_SIMP</var>}
</code></p>
</div>
</div>
<div data-if="ANS !== '\\sin^2\\theta' && ANS !== '\\cos^2\\theta' && ANS !== '1'">
This is <code><var>ANS_SIMP</var> = <var>ANS</var></code>.
</div>
<div data-else="">
This is <code><var>ANS</var></code>.
</div>
</div>
</div>
</div>
</div>
</body>
</html>