Tutorial: Trigonometric functions, models, and regression

Go to Part A: Modeling with the sine function

This tutorial: Part B: The six trigonometric functions

(This topic is also in Section 16.1 in Finite Mathematics and Applied Calculus)

The cosine function

In the "bicycle wheel definition of sine" in %%prevtut, we saw that the sine $\sin(t)$ of $t$ was defined as the $y$-coordinate of a marker on the wheel rotated ounterclockwise $t$ units. The cosine, $\cos(t)$, of $t$ is defined in almost the same way, except that this time, we use the $x$-coordinate of the marker on the wheel instead of the $y$-coordinate:

The cosine function

The cosine of a real number $t$ is given by the $x$-coordinate of the point $P$ in the following diagram, in which $t$ is the length of the arc shown.

#[Fundamental relationship between sine and cosine][Relación fundamental entre seno y coseno]#

#[Recall that the $y$-coordinate of $P$ is $\sin(t)$, so the coordinates of the point $P$ in the diagram are $(\cos(t), \sin(t))$. As its distance from the origin is $1$, the distance formula gives us][Recuerda que la coordenada $y$ de $P$ es $\sin(t)$, por lo que las coordenadas del punto $P$ en el diagrama son $(\cos(t), \sin(t))$. Como su distancia al origen es $1$, la fórmula de la distancia nos da]#

#[Square of the distance from $P$ to $(0,0)$][Cuadrado de la distancia de $P$ a $(0,0)$]# \t ${}= 1$ \\ $\left(\cos(t)\right)^2 + \left(\sin(t)\right)^2$ \t ${}= 1$

#[which we write as][que escribimos como]#

$\cos^2(t) + \sin^2(t) = 1 \qquad$ \t #[Fundamental trigonometric identity][Identidad trigonométrica fundamental]#

Graph of the conside function:

As you might expect, the graph of $y = \cos(t)$ has the same cyclical shape as that of $y = \sin(t)$. The only difference between the two is a "phase shift" (see the figure).

y = cos(t)

#[The graph of $y = \sin(t)$ is shown in a lighter shade.][La gráfica de $y = \sin(t)$ se muestra en un tono más claro.]#

#[Notes][Notas]#

As we see in the graph, the cosine function is just a shifted sine function: its graph can be obtained from that of sine by shifting it to the left $\pi/2$ units:, the graph of the sine can be obtained by shifting the graph of cosine to the right by $\pi/2$, so
$\displaystyle \cos(t) = \sin\left(t+\frac{\pi}{2}\right)$.
#[In other words, the cosine function is just a generalized sine function with the same amplitude ($A = 1$), angular velocity ($\omega = 1$) and period ($P = 2\pi$) as $\sin(t)$.][En otras palabras, la función coseno es solo una función seno generalizada con la misma amplitud ($A = 1$), velocidad angular ($\omega = 1$) y período ($P = 2\pi$) que $\sin(t)$.]#
Likewise, the graph of the sine can be obtained by shifting the graph of cosine to the right $\pi/2$ units, so
$\displaystyle \sin(t) = \cos\left(t-\frac{\pi}{2}\right)$.
The graph of $\cos(t)$ is symmetric about the $y$-axis; replacing $t$ by $-t$ gives the same value for $\cos(t)$:
$\displaystyle \cos(-t) = \cos(t)$.
By comparison, we saw that the graph of $\sin(t)$ is antisymmetric about the $y$-axis; replacing $t$ by $-t$ results in a sign change:
$\displaystyle \sin(-t) = -\sin(t)$.

#[Examples][Ejemplos]#

When $t = 0$ or $2\pi$, the point $P$ is on the $x$-axis with $x$-coordinate $1$, so that

$\cos(0) = \cos(2\pi) = 1$ \gap[20] \t $2\pi$ #[constitutes a complete revolution, or 360°.][constituye una completa revolución, o 360°.]#

When the point $P$ has moved a distance of $\dfrac{\pi}{2}$ in either direction and so makes an angle of 90° with the positive $x$-axis, its $x$-coordinate is 0, so

$\displaystyle \cos\left(\pm\frac{\pi}{2}\right)= 0$ \gap[20] \t $\dfrac{\pi}{2}$ is the radian measure of 90°.

Some for you

Modeling cosine: generalized cosine functions

As we saw in %%prevtut, we can model any sinunsoidal behavior using a generalized sine function, so, strictly speaking, there is no need to use a generlaised cosine function at all. However, there are good mathematical reasons for including cosine in our compendium of modeling functions, and it is also often more convenient to use a consine function to model a particular situation.

Generalized cosine function

#[A generalized cosine function has the following form.][Una función coseno generalizada tiene la siguiente forma.]#

$f(x) = A\cos\left[\omega(x-\beta)\right] + C$ \t \gap[40] #[Yes, we are calling the phase shift $\beta$ here as it corresponds to what we previously called $\beta$ in the graph of the generalized sine function.][Sí, estamos llamando al cambio de fase $\beta$ aquí ya que corresponde a lo que previamente llamamos $\beta$ en el gráfico de la función seno generalizada.]#

#[Thus,][Por lo tanto,]#

$A = {}$ #[amplitude][amplitud]#
$\omega = {}$ #[angular frequency][frecuencia angular]#
$\beta = {}$ #[phase shift (horizontal offset; the graph first reaches a maximum $\beta$ units to the right of the $y$-axis).][cambio de fase (desplazamiento horizontal; la gráfica primero alcanza un máximo $\beta$ unidades a la derecha del eje $y$).]#
$C = {}$ #[vertical offset (the graph is moved $C$ units up).][desplazamiento vertical (la gráfica se mueve $C$ unidades hacia arriba).]#

Here is the graph of this function, which is identical to the graph of the generlaized sine function! (Can you now see why we call the phase shift for the generalized cosine function $\beta$ instead of $\alpha$?)

#[Graph of ][Gráfica de ]# $\bold{f(x) = A\cos\left[\omega(x-\beta)\right] + C = A\sin\left[\omega(x-\alpha)\right] + C}$

$\displaystyle A = \frac{\text{highest value} - \text{lowest value}}{2} \qquad \omega = \frac{2\pi}{P} \qquad \beta = \alpha + \frac{P}{4}$
#[$C = {}$ height of baseline: Average of highest and lowest values][$C = {}$ altura de la línea base: promedio de los valores alto y bajo]#

Note Increasing or decreasing $\beta$ (or $\alpha$) by the period $P$ or multiples of $P$ has no effect on the graph (and will be permitted in the interactive exercises) as we would be moving it horizontally that distance.

#[Example][Ejemplo]# In %%prevtut we modeled the average temperature in Germany with a generalized sine function. This time, let's use a generalized cosine function instead.

$\displaystyle A = \frac{\text{highest value} - \text{lowest value}}{2} \approx \frac{20 - 0}{2} = 10$ \\ $P = 12$ \\ $\displaystyle \omega = \frac{2\pi}{P} = \frac{2\pi}{12} = \frac{\pi}{6}$ \\ $\beta = {}$ #[Value of $t$ at first high point][Valor de $t$ en el primer punto alto]# ${}\approx 6$ \\ $C = {}$ #[Average of highest and lowest values][promedio de los valores alto y bajo]# $\displaystyle \approx \frac{20+0}{2} = 10$.

#[Thus, our approximate model is][Por lo tanto, nuestro modelo aproximado es]#

$f(t)$ \t $\displaystyle {} = A\cos\left[\omega(t-\beta)\right] + C$ \\ \t $\displaystyle {} =10\cos\left[\frac{\pi}{6}(t-6)\right] + 10$.

#[Compare the sine model, from %%prevtut:][Compara el modelo seno, de %%prevtut:]#

$\displaystyle f(t) = 10\sin\left[\frac{\pi}{6}(t-3)\right] + 10$.

Here is an application to modeling a real life situation.

The other trigonometric functions

#[We can take ratios and reciprocals of sine and cosine to obtain four new functions. Here they are:][Podemos usar razones y recíprocas del seno y el coseno para obtener cuatro nuevas funciones. Aqui estan:]#

Tangent, cotangent, secant, cosecant

Each of the following is a ratio or reciprocal of sine or cosine functions, and so is not defined at values of $t$ when the denominator is zero. The result is that the graphs of these functions have vertical asymptotes at those singular ("bad") values, shown in red in the graphs below.

$\displaystyle \frac{(t)}{(t)}$ #[The tangent of $t$ is defined by][La tangente de $t$ de define por]# \t

$\displaystyle \tan(t) = \frac{\sin(t)}{\cos(t)}$

\\ \t

#[Graph of][Gráfica de]# $y = \tan(t)$

\\ $\displaystyle \frac{(t)}{(t)}$ #[The cotangent of $t$ is defined by][la cotangente de $t$ de define por]# \t

$\displaystyle \cotan(t) = \frac{\cos(t)}{\sin(t)}$

\\ \t

#[Graph of][Gráfica de]# $y = \cotan(t)$

\\ $\displaystyle \frac{1}{(t)}$ #[The secant of $t$ is defined by][la secante de $t$ de define por]# \t

$\displaystyle \sec(t) = \frac{1}{\cos(t)}$

\\ \t

#[Graph of][Gráfica de]# $y = \sec(t)$

\\ $\displaystyle \frac{1}{(t)}$ #[The cosecant of $t$ is defined by][la cosecante de $t$ de define por]# \t

$\displaystyle \cosec(t) = \frac{1}{\sin(t)}$

\\ \t

#[Graph of][Gráfica de]# $y = \cosec(t)$

Examples

1. $\displaystyle \tan(0) = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0$, whereas \\ 2. $\displaystyle \tan\left(\frac{\pi}{2}\right) = \frac{\sin(\pi/2)}{\cos(\pi/2)}$ #[is not defined, as][no es definida, porque]# $\displaystyle \cos\left(\frac{\pi}{2}\right)=0$. \t #[Notice that the graph of $\tan$ has a vertical asymptote at][Observe que la gráfica de $\tan$ tiene una asíntota vertical en]# $\dfrac{\pi}{2}.$ \\ 3. $\displaystyle \tan\left(\frac{\pi}{2}\right) = \frac{\sin(\pi/2)}{\cos(\pi/2)}$ #[is not defined, as][no es definida, porque]# $\displaystyle \cos\left(\frac{\pi}{2}\right)=0.$ \\ 4. $\displaystyle \tan\left(\frac{\pi}{4}\right) = \cotan\left(\frac{\pi}{4}\right) = 1,$ #[as][porque]# $\displaystyle \sin\left(\frac{\pi}{4}\right)=cos\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}$. \\ 5. $\displaystyle \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos(\pi/3)} = \frac{1}{1/2} = 2$. \\ 6. $\displaystyle \cosec(0) = \frac{1}{\sin(0)}$ #[is not defined, as][no es definida, porque]# $\sin(0)=0,$ #[whereas][aunque]# \\ 7. $\displaystyle \sec(0) = \frac{1}{\cos(0)} = \frac{1}{1} = 1$

Now try the exercises in Section 16.1 in Finite Mathematics and Applied Calculus. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.