From 240041fe46e0b57e56d2f662fe15cb92e80a96a7 Mon Sep 17 00:00:00 2001 From: Chrissy Date: Tue, 14 Jul 2026 16:29:19 -0400 Subject: [PATCH] small fixes to address issues --- .../act-circular-sine-cosine-computing.xml | 2 +- source/previews/PA-trig-tangent.xml | 14 +++++------ source/sec-circular-sine-cosine.xml | 24 +++++++++---------- source/sec-circular-traversing.xml | 22 ++++++++--------- 4 files changed, 31 insertions(+), 31 deletions(-) diff --git a/source/activities/act-circular-sine-cosine-computing.xml b/source/activities/act-circular-sine-cosine-computing.xml index c64e9062..2f3c27eb 100755 --- a/source/activities/act-circular-sine-cosine-computing.xml +++ b/source/activities/act-circular-sine-cosine-computing.xml @@ -50,7 +50,7 @@

- The x-coordinate of the point on the unit circle generated by a central angle with one side on the positive x-axis that measures t = -3.05 radians. (With the negative radian measure, we view the angle as opening counterclockwise from its initial side on the positive x-axis.) + The x-coordinate of the point on the unit circle generated by a central angle with one side on the positive x-axis that measures t = -3.05 radians. (With the negative radian measure, we view the angle as opening clockwise from its initial side on the positive x-axis.)

diff --git a/source/previews/PA-trig-tangent.xml b/source/previews/PA-trig-tangent.xml index 5645ed6e..90320c34 100755 --- a/source/previews/PA-trig-tangent.xml +++ b/source/previews/PA-trig-tangent.xml @@ -22,9 +22,9 @@

- Without using computational device, - find the exact value of \tan(t) at the following values: t = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}. -

 @@ -87,9 +87,9 @@ Point your browser to http://gvsu.edu/s/0yO (zero-y-Oh) to find a Desmos - worksheet with data from the tangent function already input, denoted in Desmos by the function T(x) = \tan(x). + worksheet with data from the tangent function already input, denoted in Desmos by the function T(x) = \tan(x). - Click on several of the orange points to compare your exact values in (a) with the decimal values given by Desmos. + Click on several of the orange points to compare your exact values in (a) with the decimal values given by Desmos. Add one entry to the table: x = \frac{11\pi}{24}, y = T(\frac{11\pi}{24}). diff --git a/source/sec-circular-sine-cosine.xml b/source/sec-circular-sine-cosine.xml index 5d0ce7c7..a371f062 100755 --- a/source/sec-circular-sine-cosine.xml +++ b/source/sec-circular-sine-cosine.xml @@ -36,7 +36,7 @@ Introduction

- In Section, we saw how tracking the height of a point that is traversing a cirle generates a periodic function, such as in Figure. Then, in Section, we identified a collection of 16 special points on the unit circle, as seen in Figure. + In Section, we saw how tracking the height of a point that is traversing a circle generates a periodic function, such as in Figure. Then, in Section, we identified a collection of 16 special points on the unit circle, as seen in Figure.

@@ -44,7 +44,7 @@

The unit circle with 16 labeled special points.
- +

You can also use the Desmos file at http://gvsu.edu/s/0xt to review and study the special points on the unit circle.

@@ -65,7 +65,7 @@ sine function

- Given a central angle in the unit circle that measures t radians and that intersects the circle at both (1,0) and (a,b), as shown in Figure, we define the sine of t, denoted \sin(t), by the rule + Given a central angle in the unit circle that measures t radians and that intersects the circle at both (1,0) and (a,b), as shown in Figure, we define the sine of t, denoted \sin(t), by the rule \sin(t) = b . @@ -78,7 +78,7 @@

The definition of the sine of an angle t.

- +

@@ -129,7 +129,7 @@ \frac{5\pi}{3} \frac{7\pi}{4} \frac{11\pi}{6} - 2\pi + 2\pi \sin(t) @@ -171,7 +171,7 @@ cosine function

- Given a central angle in the unit circle that measures t radians and that intersects the circle at both (1,0) and (a,b), as shown in Figure, we define the cosine of t, denoted \cos(t), by the rule + Given a central angle in the unit circle that measures t radians and that intersects the circle at both (1,0) and (a,b), as shown in Figure, we define the cosine of t, denoted \cos(t), by the rule \cos(t) = a . @@ -184,7 +184,7 @@

The definition of the cosine of an angle t.

- +

@@ -201,7 +201,7 @@ Properties of the sine and cosine functions

- Because the sine function results from tracking the y-coordinate of a point traversing the unit circle and the cosine function from the x-coordinate, the two functions have several shared properties of circular functions. + Because the sine function results from tracking the y-coordinate of a point traversing the unit circle and the cosine function from the x-coordinate, the two functions have several shared properties of circular functions.

@@ -240,8 +240,8 @@ the period of the function is p = 2\pi.

- -

+ +

@@ -280,10 +280,10 @@

-

+

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

- + diff --git a/source/sec-circular-traversing.xml b/source/sec-circular-traversing.xml index 47c234ab..a28965ca 100755 --- a/source/sec-circular-traversing.xml +++ b/source/sec-circular-traversing.xml @@ -36,13 +36,13 @@ Introduction

- 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, h, above the ground and how your height changes in tandem with the distance, d, that you have traveled around the wheel. In Figure we see a snapshot of this situation, which is available as a full animationUsed with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved. at http://gvsu.edu/s/0Dt. + 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, h, above the ground and how your height changes in tandem with the distance, d, that you have traveled around the wheel. In Figure we see a snapshot of this situation, which is available as a full animationUsed with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved. here.

- A snapshot of the motion of a cab moving around a ferris wheel. + 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. -

A snapshot of the motion of a cab moving around a ferris wheel. +

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.
@@ -69,7 +69,7 @@

- Note that we know the exact heights of certain points. Since the circle has circumference C = 24, we know that 24 = 2\pi r and therefore r = \frac{12}{\pi} \approx 3.82. Hence, the point where d = 6 (located 1/4 of the way along the circle) is at a height of h = \frac{12}{\pi} \approx 3.82. Doubling this value, the point where d = 12 has height h = \frac{24}{\pi} \approx 7.64. Other heights, such as those that correspond to d = 3 and d = 15 (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 Figure as h \approx 1.1 (for d = 3) and h \approx 6.5 (for d = 15). Using all of these observations along with the symmetry of the circle, we can determine the other entries in Table. + Note that we know the exact heights of certain points. Since the circle has circumference C = 24, we know that 24 = 2\pi r and therefore r = \frac{12}{\pi} \approx 3.82. Hence, the point where d = 6 (located 1/4 of the way along the circle) is at a height of h = \frac{12}{\pi} \approx 3.82. Doubling this value, the point where d = 12 has height h = \frac{24}{\pi} \approx 7.64. Other heights, such as those that correspond to d = 3 and d = 15 (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 Figure as h \approx 1.1 (for d = 3) and h \approx 6.5 (for d = 15). Using all of these observations along with the symmetry of the circle, we can determine the other entries in Table.

@@ -100,7 +100,7 @@ 3.821.10 - +
@@ -136,9 +136,9 @@ 3.82 1.1 0 - + - +

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 h = f(d). Using the data from the two tables and connecting the points in an intuitive way, we get the graph shown in Figure. @@ -151,7 +151,7 @@

- The function h = f(d) we have been discussing is an example of what we will call a circular function. circular function Indeed, it is apparent that if we + The function h = f(d) we have been discussing is an example of what we will call a circular function. circular function Indeed, it is apparent that if we

  • @@ -202,7 +202,7 @@

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

    @@ -262,7 +262,7 @@

    First, in Figure, we consider points P, Q, and R where Q results from traversing 1/8 of the circumference from P, and R 1/8 of the circumference from Q. In particular, we note that the distance d_1 along the circle from P to Q is the same as the distance d_2 along the circle from Q to R, and thus d_1 = d_2. At the same time, it is apparent from the geometry of the circle that the change in height h_1 from P to Q is greater than the change in height h_2 from Q to R, so h_1 \gt h_2. Thus, we can say that - 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]} .

    @@ -284,7 +284,7 @@ AV_{[P,Q]} \approx 1 \text{ and } AV_{[R,S]} \approx 0 . - 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.