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quarto/precalc/trig_functions.qmd
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@@ -30,6 +30,7 @@ We measure angles in radians, where $360$ degrees is $2\pi$ radians. By proporti
For a right triangle with angles $\theta$, $\pi/2 - \theta$, and $\pi/2$ ($0 < \theta < \pi/2$) we call the side opposite $\theta$ the "opposite" side, the shorter adjacent side the "adjacent" side and the longer adjacent side the hypotenuse.
::: {#fig-right-triangle-soh-cah-toa}
```{julia}
#| hide: true
@@ -67,10 +68,12 @@ plotly()
nothing
```
With these, the basic definitions for the primary trigonometric functions are
Labeling of a right triangle lengths in relation to a designated angle, $\theta$.
:::
::: {.callout-note icon=false}
## Trigonometric definitions
With the labelings in @fig-right-triangle-soh-cah-toa, the basic definitions for the primary trigonometric functions are
::: {.definition title="Trigonometric definitions from a right triangle"}
$$
\begin{align*}
\sin(\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} &\quad(\text{the sine function})\\
@@ -78,20 +81,16 @@ $$
\tan(\theta) &= \frac{\text{opposite}}{\text{adjacent}} &\quad(\text{the tangent function})
\end{align*}
$$
:::
:::{.callout-note}
## Note
Many students remember these through [SOH-CAH-TOA](http://mathworld.wolfram.com/SOHCAHTOA.html).
:::
Some algebra shows that $\tan(\theta) = \sin(\theta)/\cos(\theta)$. There are also $3$ reciprocal functions, the cosecant, secant and cotangent.
These definitions in terms of sides only apply for $0 \leq \theta \leq \pi/2$. More generally, if we relate any angle taken in the counter clockwise direction for the $x$-axis with a point $(x,y)$ on the *unit* circle, then we can extend these definitions - the point $(x,y)$ is also $(\cos(\theta), \sin(\theta))$.
These definitions in terms of sides only apply for $0 \leq \theta \leq \pi/2$. More generally, if we relate any angle taken in the counter clockwise direction for the $x$-axis with a point $(x,y)$ on the *unit* circle, then we can extend these definitions - the point $(x,y)$ is also $(\cos(\theta), \sin(\theta))$, cf. @fig-trig-values-along-unit-circle.
::: {#fig-trig-values-along-unit-circle}
```{julia}
#| hold: true
#| echo: false
@@ -136,38 +135,46 @@ end
imgfile = tempname() * ".gif"
gif(anim, imgfile, fps = 1)
caption = "An angle in radian measure corresponds to a point on the unit circle, whose coordinates define the sine and cosine of the angle. That is ``(x,y) = (\\cos(\\theta), \\sin(\\theta))``."
caption = ""
plotly()
ImageFile(imgfile, caption)
```
An angle in radian measure corresponds to a point on the unit circle, whose coordinates define the sine and cosine of the angle. That is $(x,y) = (\cos(\theta), \sin(\theta))$.
:::
### The trigonometric functions in Julia
Julia has the $6$ basic trigonometric functions defined through the functions `sin`, `cos`, `tan`, `csc`, `sec`, and `cot`.
Two right triangles - the one with equal, $\pi/4$, angles; and the one with angles $\pi/6$ and $\pi/3$ can have the ratio of their sides computed from basic geometry. In particular, this leads to the following values, which are usually committed to memory:
Two right triangles - the one with equal, $\pi/4$, angles; and the one with angles $\pi/6$ and $\pi/3$ can have the ratio of their sides computed from basic geometry. In particular, this leads to @tbl-basic-sin-cosine-values, values which are usually committed to memory:
$$
\begin{align*}
\sin(0) &= 0, \quad \sin(\pi/6) = \frac{1}{2}, \quad \sin(\pi/4) = \frac{\sqrt{2}}{2}, \quad\sin(\pi/3) = \frac{\sqrt{3}}{2},\text{ and } \sin(\pi/2) = 1\\
\cos(0) &= 1, \quad \cos(\pi/6) = \frac{\sqrt{3}}{2}, \quad \cos(\pi/4) = \frac{\sqrt{2}}{2}, \quad\cos(\pi/3) = \frac{1}{2},\text{ and } \cos(\pi/2) = 0.
\end{align*}
$$
::: {#tbl-basic-sin-cosine-values .striped .hover}
| $\theta$ | $\sin(\theta)$ | $\cos(\theta)$ |
|:---------:|:--------------:|:--------------:|
| $0$ | $0$ | $1$ |
| $\pi/6$ | $1/2$ | $\sqrt{3}/2$ |
| $\pi/4$ | $\sqrt{2}/2$ | $\sqrt{2}/2$ |
| $\pi/3$ | $\sqrt{3}/2$ | $1/2$ |
| $\pi/2$ | $1$ | $0$ |
: Table of sine and cosine values that can be derived from different triangles
:::
Using the circle definition allows these basic values to inform us of values throughout the unit circle.
Using the circle definition allows these basic values to inform us of values throughout the unit circle:
These all follow from the definition involving the unit circle:
* If the angle $\theta$ corresponds to a point $(x,y)$ on the unit circle, then the angle $-\theta$ corresponds to $(x, -y)$. So $\sin(\theta) = - \sin(-\theta)$ (an odd function), but $\cos(\theta) = \cos(-\theta)$ (an even function).
* If the angle $\theta$ corresponds to a point $(x,y)$ on the unit circle, then rotating by $\pi$ moves the points to $(-x, -y)$. So $x = \cos(\theta) = - \cos(\theta + \pi)$, and $y = \sin(\theta) = -\sin(\theta + \pi)$.
* If the angle $\theta$ corresponds to a point $(x,y)$ on the unit circle, then the angle $-\theta$ corresponds to $(x, -y)$. So $\sin(\theta) = - \sin(-\theta)$ (an odd function), but $\cos(\theta) = \cos(-\theta)$ (an even function).
* If the angle $\theta$ corresponds to a point $(x,y)$ on the unit circle, then rotating by $\pi$ moves the points to $(-x, -y)$. So $\cos(\theta) = x = - \cos(\theta + \pi)$, and $\sin(\theta) = y = -\sin(\theta + \pi)$.
* If the angle $\theta$ corresponds to a point $(x,y)$ on the unit circle, then rotating by $\pi/2$ moves the points to $(-y, x)$. So $\cos(\theta) = x = \sin(\theta + \pi/2)$.
* If the angle $\theta$ corresponds to a point $(x,y)$ on the unit circle, then rotating by $\pi/2$ moves the points to $(-y, x)$. So $x = \cos(\theta) = \sin(\theta + \pi/2)$.
The fact that $x^2 + y^2 = 1$ for the unit circle leads to the "Pythagorean identity" for trigonometric functions:
@@ -232,7 +239,7 @@ opposite = adjacent * tan(theta)
Having some means to compute an angle and then a tangent of that angle handy is not a given, so the linked to article provides a few other methods taking advantage of similar triangles.
You can also measure distance with your [thumb](http://www.vendian.org/mncharity/dir3/bodyruler_angle/) or fist. How? The fist takes up about $10$ degrees of view when held straight out. So, pacing off backwards until the fist completely occludes the tree will give the distance of the adjacent side of a right triangle. If that distance is $30$ paces what is the height of the tree? Well, we need some facts. Suppose your pace is $3$ feet. Then the adjacent length is $90$ feet. The multiplier is the tangent of $10$ degrees, or:
You can also measure distance with your [thumb](http://www.vendian.org/mncharity/dir3/bodyruler_angle/) or fist. How? The fist takes up about $10$ degrees of view when held straight out. So, pacing off backwards until the fist completely occludes the tree can give the distance of the adjacent side of a right triangle. If that distance is $30$ paces what is the height of the tree? Well, we need some facts. Suppose your pace is $3$ feet. Then the adjacent length is $90$ feet. The multiplier is the tangent of $10$ degrees, or:
```{julia}
@@ -259,40 +266,52 @@ This could be reversed. If you know the height of something a distance away that
### Basic properties
The sine function is defined for all real $\theta$ and has a range of $[-1,1]$. Clearly as $\theta$ winds around the $x$-axis, the position of the $y$ coordinate begins to repeat itself. We say the sine function is *periodic* with period $2\pi$. A graph will illustrate:
The sine function is defined for all real $\theta$ and has a range of $[-1,1]$. Clearly as $\theta$ winds around the $x$-axis, the position of the $y$ coordinate begins to repeat itself. We say the sine function is *periodic* with period $2\pi$. @fig-sin-over-0-4-pi illustrates. The graph shows two periods. The wavy aspect of the graph is why this function is used to model periodic motions, such as the amount of sunlight in a day, or the alternating current powering a computer.
::: {#fig-sin-over-0-4-pi}
```{julia}
#| echo: false
plot(sin, 0, 4pi)
```
The graph shows two periods. The wavy aspect of the graph is why this function is used to model periodic motions, such as the amount of sunlight in a day, or the alternating current powering a computer.
Graph of $f(x) = \sin(x)$ over $[0, 4\pi]$
:::
From this graph---or considering when the $y$ coordinate is $0$---we see that the sine function has zeros at any integer multiple of $\pi$, or $k\pi$, $k$ in $\dots,-2,-1, 0, 1, 2, \dots$.
From the graph of $\sin(x)$---or considering when the $y$ coordinate is $0$ on the unit circle---we see that the sine function has zeros at any integer multiple of $\pi$, or $k\pi$, $k$ in $\dots,-2,-1, 0, 1, 2, \dots$.
The cosine function is similar, in that it has the same domain and range, but is "out of phase" with the sine curve. A graph of both shows the two are related:
The cosine function is similar, in that it has the same domain and range, but is "out of phase" with the sine curve. @fig-sin-cosine-graph illustrates.
::: {#fig-sin-cosine-graph}
```{julia}
#| echo: false
plot(sin, 0, 4pi, label="sin")
plot!(cos, 0, 4pi, label="cos")
```
: Graph of $\sin(x)$ and $\cos(x)$ over $[0, 4\pi]$. The cosine graph lags the sine graph by $\pi/2$, or $\cos(x) = \sin(x + \pi/2)$.
:::
The cosine function is just a shift of the sine function (or vice versa). We see that the zeros of the cosine function happen at points of the form $\pi/2 + k\pi$, $k$ in $\dots,-2,-1, 0, 1, 2, \dots.$
The tangent function does not have all $\theta$ for its domain, rather those points where division by $0$ occurs are excluded. These occur when the cosine is $0$, or, again, at $\pi/2 + k\pi$, $k$ in $\dots,-2,-1, 0, 1, 2, \dots.$ The range of the tangent function will be all real $y$.
The tangent function does not have all $\theta$ for its domain, rather those points where division by $0$ occurs are excluded; when the cosine is $0$, or, again, at $\pi/2 + k\pi$, $k$ in $\dots,-2,-1, 0, 1, 2, \dots.$ The range of the tangent function will be all real $y$.
The tangent function is also periodic, but not with period $2\pi$, but rather just $\pi$. A graph will show this. Here we avoid the vertical asymptotes using `rangeclamp`:
The tangent function is also periodic, but not with period $2\pi$, but rather just $\pi$. @fig-graph-of-tangent-minus10-10 illustrates.
::: {#fig-graph-of-tangent-minus10-10}
```{julia}
#| echo: false
plot(rangeclamp(tan), -10, 10, label="tan")
```
Graph of $f(x) = \tan(x)$ over $[-10, 10]$ showing the periodic nature of the function. The vertical asymptotes were avoided by plotting `rangeclamp(tan)`.
:::
##### Example sums of sines
@@ -303,34 +322,43 @@ $$
g(x) = a + b \sin((2\pi n)x)
$$
That is a graph of $g$ will be the sine curve shifted up by $a$ units, scaled vertically by $b$ units and has a period of $1/n$. We see a simple plot here where we can verify the transformation:
That is a graph of $g$ will be the sine curve shifted up by $a$ units, scaled vertically by $b$ units and has a period of $1/n$.
@fig-plot-of-2sin-2pi-n-x verifies the transformation:
::: {#fig-plot-of-2sin-2pi-n-x}
```{julia}
g(x; b=1, n=1) = b*sin(2pi*n*x)
g1(x) = 1 + g(x, b=2, n=3)
plot(g1, 0, 1)
plot(g1, 0, 1; xticks = (0:1/3:1, ["0", "1/3", "2/3", "1"]))
```
Plot of $g(x) = a + b \sin((2\pi n)x)$ with $n=1/3$ so the period of the sinusoical function is $1/3
:::
We can consider the sum of such functions, for example
@fig-sum-two-gs shows the sum of two such functions. Though still periodic, we can see with this simple example that sums of different sine functions can have somewhat complicated graphs.
::: {#fig-sum-two-gs}
```{julia}
g2(x) = 1 + g(x, b=2, n=3) + g(x, b=4, n=5)
plot(g2, 0, 1)
```
Though still periodic, we can see with this simple example that sums of different sine functions can have somewhat complicated graphs.
Graph of sum of two sine functions, one with period $1/3$ one with period $1/5$. The resulting function has period $1$.
:::
Sine functions can be viewed as the `x` position of a point traveling around a circle so `g(x, b=2, n=3)` is the `x` position of point traveling around a circle of radius $2$ that completes a circuit in $1/3$ units of time.
The superposition of the two sine functions that `g2` represents could be viewed as the position of a circle moving around a point that is moving around another circle. The following graphic, with $b_1=1/3, n_1=3, b_2=1/4$, and $n_2=4$, shows an example that produces the related cosine sum (moving right along the $x$ axis), the sine sum (moving down along the $y$ axis, *and* the trace of the position of the point generating these two plots.
The superposition of the two sine functions that `g2` represents could be viewed as the position of a circle moving around a point that is moving around another circle. @fig-superposition-of-sines-cosines, with $b_1=1/3, n_1=3, b_2=1/4$, and $n_2=4$, shows an example that produces the related cosine sum (moving right along the $x$ axis), the sine sum (moving down along the $y$ axis, *and* the trace of the position of the point generating these two plots.
::: {#fig-superposition-of-sines-cosines}
```{julia}
#| hold: true
#| echo: false
#| cache: true
gr()
@@ -390,11 +418,14 @@ end
imgfile = tempname() * ".gif"
gif(anim, imgfile, fps = 5)
caption = "Superposition of sines and cosines represented by an epicycle"
caption = ""
plotly()
ImageFile(imgfile, caption)
```
Superposition of sines and cosines represented by an epicycle.
:::
As can be seen, even a somewhat simple combination can produce complicated graphs (a fact known to [Ptolemy](https://en.wikipedia.org/wiki/Deferent_and_epicycle)) . How complicated can such a graph get? This won't be answered here, but for fun enjoy this video produced by the same technique using more moving parts from the [`Javis.jl`](https://github.com/Wikunia/Javis.jl/blob/master/examples/fourier.jl) package:
@@ -431,6 +462,7 @@ More generally, suppose we have two angles $\alpha$ and $\beta$, can we represen
Suppose both $\alpha$ and $\beta$ are positive with $\alpha + \beta \leq \pi/2$. Then using right triangle geometry we can associate the sine and cosine of $\alpha + \beta$ with distances in this figure:
::: {#fig-sin-cos-alpha-plus-beta-and-beta}
```{julia}
#| echo: false
gr()
@@ -534,25 +566,25 @@ plot!(Shape([F,B]), fill=(:black, 0.35))
annotate!(map(s ->getindex(txtpoints,s), collect(keys(txtpoints))))
p1
plot(p1, p2)
```
Another right triangle with hypotenuse of length $1$ can be made by isolating the angle $\beta$, as below:
The left figure labels the sides of a right triangle with angle $\alpha + \beta$, the right figure labels the sides of a right triangle with angle $\beta$.
:::
```{julia}
#| echo: false
p2
```
In @fig-cos-alpha-beta-sin-alpha-beta we make two more right triangles one with hypotenuse $\cos(\beta)$ and one with hypotenuse $\sin(\beta)$; each having an angle $\alpha$, the latter using some geometry, for which we can apply right-triangle trigonometry to find the length of their respective sides.
We can make two more right triangles one with hypotenuse $\cos(\beta)$ and one with hypotenuse $\sin(\beta)$; each having an angle $\alpha$, the latter using some geometry, for which we can apply right-triangle trigonometry to find the length of their sides.
::: {#fig-cos-alpha-beta-sin-alpha-beta}
```{julia}
#| echo: false
plot(p3, p4)
```
From the left figure and the initial triangle, by comparing the lengths along the $x$ direction, we can see the decomposition:
Two triangles with angle $\alpha$ and hypotenuses $\cos(\beta)$ and $\sin(\beta)$.
:::
From the left side of #fig-cos-alpha-beta-sin-alpha-beta and the initial triangle, by comparing the lengths along the $x$ direction, we can see the decomposition:
$$
\cos(\alpha)\cos(\beta) = \cos(\alpha + \beta) + \sin(\alpha)\sin(\beta)
@@ -564,10 +596,9 @@ $$
\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)
$$
These lead to:
All combined, these lead to:
::: {.callout-note icon=false}
## The *sum* formulas for sine and cosine
::: {.relationship title="The sum formulas for sine and cosine"}
$$
\begin{align*}
@@ -575,12 +606,12 @@ $$
\cos(\alpha + \beta) &= \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)
\end{align*}
$$
:::
Taking $\alpha = \beta$ we immediately get
::: {.callout-note icon=false}
## The "double-angle" formulas
::: {.relationship title="The double-angle formulas"}
$$
\begin{align*}
\sin(2\alpha) &= 2\sin(\alpha)\cos(\alpha)\\
@@ -589,17 +620,16 @@ $$
$$
:::
The latter looks like the Pythagorean identify, but has a minus sign. In fact, the Pythagorean identify is often used to rewrite this, for example $\cos(2\alpha) = 2\cos^2(\alpha) - 1$ or $1 - 2\sin^2(\alpha)$.
The latter looks like the Pythagorean identify, but has a minus sign. In fact, the Pythagorean identify is often used to rewrite this formula, for example $\cos(2\alpha) = 2\cos^2(\alpha) - 1$ or $1 - 2\sin^2(\alpha)$.
Applying the above with $\alpha = \beta/2$, we get that $\cos(\beta) = 2\cos^2(\beta/2) -1$. Similarly, using the Pythagorean identity a formula for sine can be done; when rearranged these yield the "half-angle" formulas:
Applying the above with $\alpha = \beta/2$, we get that $\cos(\beta) = 2\cos^2(\beta/2) -1$. Similarly, using the Pythagorean identity a formula for sine can be identified; when rearranged these yield the "half-angle" formulas:
::: {.callout-note icon=false}
## The "half-angle" formula
::: {.relationship title="The half-angle formulas"}
$$
\begin{align*}
\sin^2(\frac{\beta}{2}) &= \frac{1 - \cos(\beta)}{2}\\
\cos^2(\frac{\beta}{2}) &= \frac{1 + \cos(\beta)}{2}
\sin^2\left(\frac{\beta}{2}\right) &= \frac{1 - \cos(\beta)}{2}\\
\cos^2\left(\frac{\beta}{2}\right) &= \frac{1 + \cos(\beta)}{2}
\end{align*}
$$
:::
@@ -633,6 +663,13 @@ $$
That is the angle for a multiple of $n+1$ can be expressed in terms of the angle with a multiple of $n$ and $n-1$. This can be used recursively to find expressions for $\cos(n\theta)$ in terms of polynomials in $\cos(\theta)$.
For example,
```{julia}
@syms θ
sympy.expand_trig(cos(5 * θ))
```
## Inverse trigonometric functions
@@ -640,29 +677,31 @@ That is the angle for a multiple of $n+1$ can be expressed in terms of the angle
The trigonometric functions are all periodic. In particular they are not monotonic over their entire domain. This means there is no *inverse* function applicable. However, by restricting the domain to where the functions are monotonic, inverse functions can be defined:
* For $\sin(x)$, the restricted domain of $[-\pi/2, \pi/2]$ allows for the arcsine function to be defined. In `Julia` this is implemented with `asin`.
* For $\cos(x)$, the restricted domain of $[0,\pi]$ allows for the arccosine function to be defined. In `Julia` this is implemented with `acos`.
* For $\tan(x)$, the restricted domain of $(-\pi/2, \pi/2)$ allows for the arctangent function to be defined. In `Julia` this is implemented with `atan`.
* For $\sin(x)$, the restricted domain of $[-\pi/2, \pi/2]$ allows for the arcsine function to be defined. In `Julia` this is implemented with `asin`.
* For $\cos(x)$, the restricted domain of $[0,\pi]$ allows for the arccosine function to be defined. In `Julia` this is implemented with `acos`.
* For $\tan(x)$, the restricted domain of $(-\pi/2, \pi/2)$ allows for the arctangent function to be defined. In `Julia` this is implemented with `atan`.
For example, the arcsine function is defined for $-1 \leq x \leq 1$ and has a range of $-\pi/2$ to $\pi/2$:
::: {#fig-arcsin-arctan}
```{julia}
plot(asin, -1, 1)
#| echo: false
p1 = plot(asin, -1, 1; legend=false, title="arcsin")
p2 = plot(atan, -15, 15; legend=false, title="arctan")
plot(p1, p2)
```
The arctangent has domain of all real $x$. It has shape given by:
The function $f(x) = \arcsin(x)$ has domain $[-1,1]$, whereas the function $f(x) = \arctan(x)$ has domain $(-\infty, \infty)$.
:::
```{julia}
plot(atan, -10, 10)
```
The horizontal asymptotes are $y=\pi/2$ and $y=-\pi/2$.
### Implications of a restricted domain
##### Example: Implications of a restricted domain
Notice that $\sin(\arcsin(x)) = x$ for any $x$ in $[-1,1]$, but, of course, not for all $x$, as the output of the sine function can't be arbitrarily large.
@@ -805,7 +844,7 @@ $$
Both $\theta_0$ and $\theta_1$ are measured with respect to the coordinate system that looks like the $x-y$ plane. The red coordinate system is used to identify the angle of incidence for the second bending. Some right-triangle geometry relates the new angle $\theta'_1$ with $\theta_1$ through $\theta'_1 = \alpha - \theta_1$. With this new angle of incidence, the angle of refraction, $\theta'_2$, satisfies:
$$
n1 \sin(\theta'_1) = n2 \sin(\theta'_2)
n_1 \sin(\theta'_1) = n_2 \sin(\theta'_2)
$$
Or
@@ -868,9 +907,9 @@ $$
d = \pi + 2i - 4 \arcsin(\frac{1}{n} \sin(i)).
$$
Graphing this for incident angles between $0$ and $\pi/2$ we have:
@fig-plot-of-deflection-for-different-incident-angles shows the deflection for incident angles between $0$ and $\pi/2$
::: {#fig-plot-of-deflection-for-different-incident-angles}
```{julia}
#| hold: true
n = 4/3
@@ -878,6 +917,9 @@ d(i) = pi + 2i - 4 * asin(sin(i)/n)
plot(d, 0, pi/2)
```
Plot of deflection for different incident angles
:::
Descartes was interested in the minimum value of this graph, as it relates to where the light concentrates. This is roughly at $1$ radian or about $57$ degrees:
@@ -915,9 +957,9 @@ A few things become clear from the above two representations:
* Using the initial definition, we see that the zeros of $T_n(x)$ all occur within $[-1,1]$ and happen when $n\arccos(x) = k\pi + \pi/2$, or $x=\cos((2k+1)/n \cdot \pi/2)$ for $k=0, 1, \dots, n-1$.
Other properties of this polynomial family are not at all obvious. One is that amongst all polynomials of degree $n$ with roots in $[-1,1]$, $T_n(x)$ will be the smallest in magnitude (after we divide by the leading coefficient to make all polynomials considered to be monic). We check this for one case. Take $n=4$, then we have: $T_4(x) = 8x^4 - 8x^2 + 1$. Compare this with $q(x) = (x+3/5)(x+1/5)(x-1/5)(x-3/5)$ (evenly spaced zeros):
Other properties of this polynomial family are not at all obvious. One is that amongst all polynomials of degree $n$ with roots in $[-1,1]$, $T_n(x)$ will be the smallest in magnitude (after we divide by the leading coefficient to make all polynomials considered to be monic). We check this for one case. Take $n=4$, then we have: $T_4(x) = 8x^4 - 8x^2 + 1$. We compare this polynomial with $q(x) = (x+3/5)(x+1/5)(x-1/5)(x-3/5)$ (evenly spaced zeros) in @fig-plot-T4-q-showing-chebyshev-minimal.
::: {#fig-plot-T4-q-showing-chebyshev-minimal}
```{julia}
T4(x) = (8x^4 - 8x^2 + 1) / 8
q(x) = (x+3/5)*(x+1/5)*(x-1/5)*(x-3/5)
@@ -925,14 +967,17 @@ plot(abs ∘ T4, -1,1, label="|T₄|")
plot!(abs ∘ q, -1,1, label="|q|")
```
We will return to this family of polynomials in the section on Orthogonal Polynomials.
The monic Chebyshev polynomial is has the smallest maximum value of $[-1,1]$ of all monic polynomials of the same degree
:::
We will return to this family of polynomials in the section on orthogonal polynomials.
## Hyperbolic trigonometric functions
Related to the trigonometric functions are the hyperbolic trigonometric functions. Instead of associating a point $(x,y)$ on the unit circle with an angle $\theta,$ we associate a point $(x,y)$ on the unit *hyperbola* ($x^2 - y^2 = 1$). We define the hyperbolic sine ($\sinh$) and hyperbolic cosine ($\cosh$) through $(\cosh(\theta), \sinh(\theta)) = (x,y)$.
::: {#fig-hyperbolic-trig-functions-from-unit-hyperbola}
```{julia}
#| echo: false
let
@@ -940,7 +985,7 @@ let
# y^2 = x^2 - 1
top(x) = sqrt(x^2 - 1)
p = plot(; legend=false, aspect_ratio=:equal)
p = plot(; legend=false, framestyle=:origin, aspect_ratio=:equal)
x₀ = 2
xs = range(1, x₀, length=100)
@@ -976,6 +1021,8 @@ let
p
end
```
Figure showing the definitions of $\cosh(x)$ and $\sinh(x)$ using the unit hyperbola $x^2 - y^2 = 1$
:::
These values are more commonly expressed using the exponential function as:
@@ -1004,12 +1051,12 @@ What is bigger $\sin(1.23456)$ or $\cos(6.54321)$?
```{julia}
#| hold: true
#| echo: false
a = sin(1.23456) > cos(6.54321)
choices = [raw"``\sin(1.23456)``", raw"``\cos(6.54321)``"]
answ = a ? 1 : 2
radioq(choices, answ, keep_order=true)
answer = a ? 1 : 2
explanation = "Compare with `sin(1.23456) > cos(6.54321)`"
buttonq(choices, answer; explanation)
```
###### Question
@@ -1019,13 +1066,13 @@ Let $x=\pi/4$. What is bigger $\cos(x)$ or $x$?
```{julia}
#| hold: true
#| echo: false
x = pi/4
a = cos(x) > x
choices = [raw"``\cos(x)``", "``x``"]
answ = a ? 1 : 2
radioq(choices, answ, keep_order=true)
answer = a ? 1 : 2
explanation = "Compare with `cos(pi/4) > pi/4`"
radioq(choices, answer; explanation)
```
###### Question
@@ -1041,8 +1088,8 @@ choices = [
raw"``\cos(x) = \sin(x - \pi/2)``",
raw"``\cos(x) = \sin(x + \pi/2)``",
raw"``\cos(x) = \pi/2 \cdot \sin(x)``"]
answ = 2
radioq(choices, answ)
answer = 2
buttonq(choices, answer)
```
###### Question
@@ -1058,8 +1105,9 @@ choices = [
L"The values $k\pi$ for $k$ in $\dots, -2, -1, 0, 1, 2, \dots$",
L"The values $\pi/2 + k\pi$ for $k$ in $\dots, -2, -1, 0, 1, 2, \dots$",
L"The values $2k\pi$ for $k$ in $\dots, -2, -1, 0, 1, 2, \dots$"]
answ = 2
radioq(choices, answ, keep_order=true)
answer = 2
explanation = "The secant is the reciprocal of the cosine function"
buttonq(choices, answer; explanation)
```
###### Question
@@ -1117,37 +1165,40 @@ numericq(val)
The sine function is an *odd* function.
* The hyperbolic sine is:
* The hyperbolic sine is:
```{julia}
#| hold: true
#| echo: false
choices = ["odd", "even", "neither"]
answ = 1
radioq(choices, answ, keep_order=true)
answer = 1
explanation = "subsitute `-x` into the exponential formula to see"
buttonq(choices, answer; explanation)
```
* The hyperbolic cosine is:
* The hyperbolic cosine is:
```{julia}
#| hold: true
#| echo: false
choices = ["odd", "even", "neither"]
answ = 2
radioq(choices, answ, keep_order=true)
answer = 2
explanation = L"The value of $\cosh(-x)$ is the $y$ position of the point $(x,y)$ refelected through the $y$ axis, so is unchanged."
buttonq(choices, answer; explanation)
```
* The hyperbolic tangent is:
* The hyperbolic tangent is:
```{julia}
#| hold: true
#| echo: false
choices = ["odd", "even", "neither"]
answ = 1
radioq(choices, answ, keep_order=true)
answer = 1
explanation = "A ratio of an odd function and an even function is *odd*"
buttonq(choices, answer; explanation)
```
###### Question