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mattboelkins merged 1 commit into from
Jul 15, 2026
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small fixes to address issues #61

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2 changes: 1 addition & 1 deletion source/activities/act-circular-sine-cosine-computing.xml
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Expand Up @@ -50,7 +50,7 @@
<task workspace="4cm" xml:id="act-circular-sine-cosine-computing-task-2">
<statement>
<p>
The <m>x</m>-coordinate of the point on the unit circle generated by a central angle with one side on the positive <m>x</m>-axis that measures <m>t = -3.05</m> radians. (With the negative radian measure, we view the angle as opening counterclockwise from its initial side on the positive <m>x</m>-axis.)
The <m>x</m>-coordinate of the point on the unit circle generated by a central angle with one side on the positive <m>x</m>-axis that measures <m>t = -3.05</m> radians. (With the negative radian measure, we view the angle as opening clockwise from its initial side on the positive <m>x</m>-axis.)
</p>
</statement>
<hint/>
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14 changes: 7 additions & 7 deletions source/previews/PA-trig-tangent.xml
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Expand Up @@ -22,9 +22,9 @@
<task workspace="5cm" xml:id="PA-trig-tangent-task-0">
<statement>
<p>
Without using computational device,
find the exact value of <m>\tan(t)</m> at the following values: <m>t = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}</m>.
<!--
Without using a computational device,
find the exact value of <m>\tan(t)</m> at the following values: <m>t = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}</m>.
<!--

<sidebyside>
<tabular>
Expand All @@ -45,10 +45,10 @@
<mrow>\tan \left( \frac{2\pi}{3} \right) =</mrow>
<mrow>\tan \left( \frac{3\pi}{4} \right) =</mrow>
<mrow>\tan \left( \frac{5\pi}{6} \right) =</mrow>
</md>
</md>
</cell>
</row>
</tabular>
</tabular>
</sidebyside>
--></p>
</statement>
Expand Down Expand Up @@ -87,9 +87,9 @@
Point your browser to
<url href="http://gvsu.edu/s/0yO">http://gvsu.edu/s/0yO</url>
(<q>zero-y-Oh</q>) to find a <em>Desmos</em>
worksheet with data from the tangent function already input, denoted in <em>Desmos</em> by the function <m>T(x) = \tan(x)</m>.
worksheet with data from the tangent function already input, denoted in <em>Desmos</em> by the function <m>T(x) = \tan(x)</m>.

Click on several of the orange points to compare your exact values in (a) with the decimal values given by <em>Desmos</em>.
Click on several of the orange points to compare your exact values in (a) with the decimal values given by <em>Desmos</em>.

Add one entry to the table:
<m>x = \frac{11\pi}{24}</m>, <m>y = T(\frac{11\pi}{24})</m>.
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24 changes: 12 additions & 12 deletions source/sec-circular-sine-cosine.xml
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Expand Up @@ -36,15 +36,15 @@

<subsection><title>Introduction</title>
<p>
In <xref ref="sec-circular-traversing">Section</xref>, we saw how tracking the height of a point that is traversing a cirle generates a periodic function, such as in <xref ref="F-act-circular-functions-properties-graph">Figure</xref>. Then, in <xref ref="sec-circular-unit-circle">Section</xref>, we identified a collection of <m>16</m> special points on the unit circle, as seen in <xref ref="F-circular-sine-all-16">Figure</xref>.
In <xref ref="sec-circular-traversing">Section</xref>, we saw how tracking the height of a point that is traversing a circle generates a periodic function, such as in <xref ref="F-act-circular-functions-properties-graph">Figure</xref>. Then, in <xref ref="sec-circular-unit-circle">Section</xref>, we identified a collection of <m>16</m> special points on the unit circle, as seen in <xref ref="F-circular-sine-all-16">Figure</xref>.
</p>

<figure xml:id="F-circular-sine-all-16">
<caption>The unit circle with <m>16</m> labeled special points.</caption>
<image source="images/unit-circle-16-all-labeled" width="80%"><description><p>The unit circle with <m>16</m> labeled special points.</p>
</description></image>
</figure>

<p>
You can also use the <em>Desmos</em> file at <url href="https://www.desmos.com/calculator/jgddn7tzxg">http://gvsu.edu/s/0xt</url> to review and study the special points on the unit circle.
</p>
Expand All @@ -65,7 +65,7 @@
<idx><h>sine function</h></idx>
<statement>
<p>
Given a central angle in the unit circle that measures <m>t</m> radians and that intersects the circle at both <m>(1,0)</m> and <m>(a,b)</m>, as shown in <xref ref="F-def-sine">Figure</xref>, we define the <term>sine of <m>t</m></term>, denoted <m>\sin(t)</m>, by the rule
Given a central angle in the unit circle that measures <m>t</m> radians and that intersects the circle at both <m>(1,0)</m> and <m>(a,b)</m>, as shown in <xref ref="F-def-sine">Figure</xref>, we define the <term>sine of <m>t</m></term>, denoted <m>\sin(t)</m>, by the rule
<me>
\sin(t) = b
</me>.
Expand All @@ -78,7 +78,7 @@
<image source="images/sine-defn"><description><p>The definition of the sine of an angle <m>t</m>.</p>
</description></image>
</figure>

</sidebyside>

<p>
Expand Down Expand Up @@ -129,7 +129,7 @@
<cell><m>\frac{5\pi}{3}</m></cell>
<cell><m>\frac{7\pi}{4}</m></cell>
<cell><m>\frac{11\pi}{6}</m></cell>
<cell><m>2\pi</m></cell>
<cell><m>2\pi</m></cell>
</row>
<row halign="center">
<cell><m>\sin(t)</m></cell>
Expand Down Expand Up @@ -171,7 +171,7 @@
<idx><h>cosine function</h></idx>
<statement>
<p>
Given a central angle in the unit circle that measures <m>t</m> radians and that intersects the circle at both <m>(1,0)</m> and <m>(a,b)</m>, as shown in <xref ref="F-def-cosine">Figure</xref>, we define the <term>cosine of <m>t</m></term>, denoted <m>\cos(t)</m>, by the rule
Given a central angle in the unit circle that measures <m>t</m> radians and that intersects the circle at both <m>(1,0)</m> and <m>(a,b)</m>, as shown in <xref ref="F-def-cosine">Figure</xref>, we define the <term>cosine of <m>t</m></term>, denoted <m>\cos(t)</m>, by the rule
<me>
\cos(t) = a
</me>.
Expand All @@ -184,7 +184,7 @@
<image source="images/sine-defn-cosine"><description><p>The definition of the cosine of an angle <m>t</m>.</p>
</description></image>
</figure>

</sidebyside>

<p>
Expand All @@ -201,7 +201,7 @@
<subsection xml:id="subsec-circular-sine-cosine-properties">
<title>Properties of the sine and cosine functions</title>
<p>
Because the sine function results from tracking the <m>y</m>-coordinate of a point traversing the unit circle and the cosine function from the <m>x</m>-coordinate, the two functions have several shared properties of circular functions.
Because the sine function results from tracking the <m>y</m>-coordinate of a point traversing the unit circle and the cosine function from the <m>x</m>-coordinate, the two functions have several shared properties of circular functions.
</p>


Expand Down Expand Up @@ -240,8 +240,8 @@
the period of the function is <m>p = 2\pi</m>.
</p>
</li>
</ul>
</p>
</ul>
</p>
</assemblage>

<p>
Expand Down Expand Up @@ -280,10 +280,10 @@
</p>
</assemblage>

<p>
<p>
There are additional trends and patterns in the two functions' graphs that we explore further in the following activity.
</p>

<xi:include href="./activities/act-circular-sine-cosine-incr-CCU.xml"/>

</subsection>
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22 changes: 11 additions & 11 deletions source/sec-circular-traversing.xml
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Expand Up @@ -36,13 +36,13 @@

<subsection><title>Introduction</title>
<p>
Certain naturally occurring phenomena eventually repeat themselves, especially when the phenomenon is somehow connected to a circle. For example, suppose that you are taking a ride on a ferris wheel and we consider your height, <m>h</m>, above the ground and how your height changes in tandem with the distance, <m>d</m>, that you have traveled around the wheel. In <xref ref="F-traversing-ferris-wheel-animation">Figure</xref> we see a snapshot of this situation, which is available as a full animation<fn>Used with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved.</fn> at <url href="http://gvsu.edu/s/0Dt">http://gvsu.edu/s/0Dt</url>.
Certain naturally occurring phenomena eventually repeat themselves, especially when the phenomenon is somehow connected to a circle. For example, suppose that you are taking a ride on a ferris wheel and we consider your height, <m>h</m>, above the ground and how your height changes in tandem with the distance, <m>d</m>, that you have traveled around the wheel. In <xref ref="F-traversing-ferris-wheel-animation">Figure</xref> we see a snapshot of this situation, which is available as a full animation<fn>Used with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved.</fn> <url href="https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Investigating-Functions-with-a-Ferris-Wheel-Distance-vs-Height/">here</url>.
</p>

<figure xml:id="F-traversing-ferris-wheel-animation">
<caption>A snapshot of the motion of a cab moving around a ferris wheel.
<caption>A snapshot of the motion of a cab moving around a ferris wheel.
Reprinted with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved.</caption>
<image width="95%" source="images/traversing-ferris-wheel-animation"><description><p>A snapshot of the motion of a cab moving around a ferris wheel.
<image width="95%" source="images/traversing-ferris-wheel-animation"><description><p>A snapshot of the motion of a cab moving around a ferris wheel.
Reprinted with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved.</p>
</description></image>
</figure>
Expand All @@ -69,7 +69,7 @@
</figure>

<p>
Note that we know the exact heights of certain points. Since the circle has circumference <m>C = 24</m>, we know that <m>24 = 2\pi r</m> and therefore <m>r = \frac{12}{\pi} \approx 3.82</m>. Hence, the point where <m>d = 6</m> (located <m>1/4</m> of the way along the circle) is at a height of <m>h = \frac{12}{\pi} \approx 3.82</m>. Doubling this value, the point where <m>d = 12</m> has height <m>h = \frac{24}{\pi} \approx 7.64</m>. Other heights, such as those that correspond to <m>d = 3</m> and <m>d = 15</m> (identified on the figure by the green line segments) are not obvious from the circle's radius, but can be estimated from the grid in <xref ref="F-traversing-first-example">Figure</xref> as <m>h \approx 1.1</m> (for <m>d = 3</m>) and <m>h \approx 6.5</m> (for <m>d = 15</m>). Using all of these observations along with the symmetry of the circle, we can determine the other entries in <xref ref="T-traversing-first-example-1">Table</xref>.
Note that we know the exact heights of certain points. Since the circle has circumference <m>C = 24</m>, we know that <m>24 = 2\pi r</m> and therefore <m>r = \frac{12}{\pi} \approx 3.82</m>. Hence, the point where <m>d = 6</m> (located <m>1/4</m> of the way along the circle) is at a height of <m>h = \frac{12}{\pi} \approx 3.82</m>. Doubling this value, the point where <m>d = 12</m> has height <m>h = \frac{24}{\pi} \approx 7.64</m>. Other heights, such as those that correspond to <m>d = 3</m> and <m>d = 15</m> (identified on the figure by the green line segments) are not obvious from the circle's radius, but can be estimated from the grid in <xref ref="F-traversing-first-example">Figure</xref> as <m>h \approx 1.1</m> (for <m>d = 3</m>) and <m>h \approx 6.5</m> (for <m>d = 15</m>). Using all of these observations along with the symmetry of the circle, we can determine the other entries in <xref ref="T-traversing-first-example-1">Table</xref>.
</p>

<table xml:id="T-traversing-first-example-1">
Expand Down Expand Up @@ -100,7 +100,7 @@
<cell><m>3.82</m></cell>
<cell><m>1.1</m></cell>
<cell><m>0</m></cell>
</row>
</row>
</tabular>
</table>

Expand Down Expand Up @@ -136,9 +136,9 @@
<cell><m>3.82</m></cell>
<cell><m>1.1</m></cell>
<cell><m>0</m></cell>
</row>
</row>
</tabular>
</table>
</table>

<p>
It is apparent that each point on the circle corresponds to one and only one height, and thus we can view the height of a point as a function of the distance the point has traversed around the circle, say <m>h = f(d)</m>. Using the data from the two tables and connecting the points in an intuitive way, we get the graph shown in <xref ref="F-traversing-first-example-graph">Figure</xref>.
Expand All @@ -151,7 +151,7 @@
</figure>

<p>
The function <m>h = f(d)</m> we have been discussing is an example of what we will call a <em>circular function</em>. <idx><h>circular function</h></idx> Indeed, it is apparent that if we
The function <m>h = f(d)</m> we have been discussing is an example of what we will call a <em>circular function</em>. <idx><h>circular function</h></idx> Indeed, it is apparent that if we
<ul>
<li>
<p>
Expand Down Expand Up @@ -202,7 +202,7 @@
</sidebyside>

<p>
We assume that the point traversing the circle starts at <m>P</m> in <xref ref="F-circular-functions-properties-circle">Figure</xref>. Its height is initially <m>y = m + a</m>, and then its height decreases to <m>y = m</m> as we traverse to <m>Q</m>. Continuing, the point's height falls to <m>y = m - a</m> at <m>R</m>, and then rises back to <m>y = m</m> at <m>S</m>, and eventually back up to <m>y = m+a</m> at the top of the circle. If we plot these heights continuously as a function of distance, <m>d</m>, traversed around the circle, we get the curve shown at right in <xref ref="F-act-circular-functions-properties-graph">Figure</xref>. This curve has several important features for which we introduce important terminology.
We assume that the point traversing the circle starts at <m>P</m> in <xref ref="F-circular-functions-properties-circle">Figure</xref>. Its height is initially <m>y = m + a</m>, and then its height decreases to <m>y = m</m> as we traverse to <m>Q</m>. Continuing, the point's height falls to <m>y = m - a</m> at <m>R</m>, and then rises back to <m>y = m</m> at <m>S</m>, and eventually back up to <m>y = m+a</m> at the top of the circle. If we plot these heights continuously as a function of distance, <m>d</m>, traversed around the circle, we get the curve shown at right in <xref ref="F-act-circular-functions-properties-graph">Figure</xref>. This curve has several important features for which we introduce important terminology.
</p>


Expand Down Expand Up @@ -262,7 +262,7 @@
<p>
First, in <xref ref="F-circular-traversing-aroc-eighth">Figure</xref>, we consider points <m>P</m>, <m>Q</m>, and <m>R</m> where <m>Q</m> results from traversing <m>1/8</m> of the circumference from <m>P</m>, and <m>R</m> <m>1/8</m> of the circumference from <m>Q</m>. In particular, we note that the distance <m>d_1</m> along the circle from <m>P</m> to <m>Q</m> is the same as the distance <m>d_2</m> along the circle from <m>Q</m> to <m>R</m>, and thus <m>d_1 = d_2</m>. At the same time, it is apparent from the geometry of the circle that the change in height <m>h_1</m> from <m>P</m> to <m>Q</m> is greater than the change in height <m>h_2</m> from <m>Q</m> to <m>R</m>, so <m>h_1 \gt h_2</m>. Thus, we can say that
<me>
AV_{[P,Q]} = \frac{h_1}{d_1} \gt \frac{h_2}{d_2} = AV_{[Q,R]}
AV_{[P,Q]} = \frac{h_1}{d_1} \gt \frac{h_2}{d_2} = AV_{[Q,R]}
</me>.
</p>

Expand All @@ -284,7 +284,7 @@
<me>
AV_{[P,Q]} \approx 1 \text{ and } AV_{[R,S]} \approx 0
</me>.
This information tells us that a circular function appears to change most rapidly for points near its midline and to change least rapidly for points near its highest and lowest values.
This information tells us that a circular function appears to change most rapidly for points near its midline and to change least rapidly for points near its highest and lowest values.
</p>

<p>
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