Trigonometric functions

We have informally used some of the trigonometric functions in examples so far. In this section we quickly review their definitions and some basic properties.

The trigonometric functions are used to describe relationships between triangles and circles as well as oscillatory motions. With such a wide range of utility it is no wonder that they pop up in many places and their origins date to Hipparcus and Ptolemy over 2000 years ago.

The 6 basic trigonometric functions

We measure angles in radians, where $360$ degrees is $2\pi$ radians. By proportions, $180$ degrees is $\pi$ radian, $90$ degrees is $\pi/2$ radians, $60$ degrees is $\pi/3$ radians, etc. In general, $x$ degrees is $2\pi \cdot x / 360$ radians.

For a right triangle with angles $\theta$, $\pi/2 - \theta$, and $\pi/2$ we call the side opposite $\theta$ the "opposite" side, the shorter adjacent side the "adjacent" side and the longer adjacent side the hypotenuse.

With these, the basic definitions for the primary trigonometric functions are

\[ ~ \begin{align} \sin(\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} &\quad(\text{the sine function})\\ \cos(\theta) &= \frac{\text{adjacent}}{\text{hypotenuse}} &\quad(\text{the cosine function})\\ \tan(\theta) &= \frac{\text{opposite}}{\text{adjacent}}. &\quad(\text{the tangent function}) \end{align} ~ \]

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 $(\cos(\theta), \sin(\theta))$.

An angle in radian measure corresponds to a point on the unit circle, whose coordinates define the sine and cosine of the angle.

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:

\[ ~ \begin{align} \sin(0) &= 0, \quad \sin(\pi/6) = \frac{1}{2}, \quad \sin(\pi/4) = \frac{\sqrt{2}}{2}, \quad\sin(\pi/3) = \frac{\sqrt{3}}{2},\text{ and } \sin(\pi/2) = 1\\ \cos(0) &= 1, \quad \cos(\pi/6) = \frac{\sqrt{3}}{2}, \quad \cos(\pi/4) = \frac{\sqrt{2}}{2}, \quad\cos(\pi/3) = \frac{1}{2},\text{ and } \cos(\pi/2) = 0. \end{align} ~ \]

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:

The fact that $x^2 + y^2 = 1$ for the unit circle leads to the "Pythagorean identity" for trigonometric functions:

\[ ~ \sin(\theta)^2 + \cos(\theta)^2 = 1. ~ \]

This basic fact can be manipulated many ways. For example, dividing through by $\cos(\theta)^2$ gives the related identity: $\tan(\theta)^2 + 1 = \sec(\theta)^2$.

Julia's functions can compute values for any angles, including these fundamental ones:

[cos(theta) for theta in [0, pi/6, pi/4, pi/3, pi/2]]
5-element Array{Float64,1}:
 1.0
 0.8660254037844387
 0.7071067811865476
 0.5000000000000001
 6.123233995736766e-17

These are floating point approximations, as can be seen clearly in the last value. Symbolic math can be used if exactness matters:

using CalculusWithJulia   # to load the `SymPy` package
using Plots
cos.([0, PI/6, PI/4, PI/3, PI/2])
\[ \left[ \begin{array}{r}1\\\frac{\sqrt{3}}{2}\\\frac{\sqrt{2}}{2}\\\frac{1}{2}\\0\end{array} \right] \]
Example

Measuring the height of a tree may be a real-world task for some, but a typical task for trigonometry students. How might it be done? If a right triangle can be formed where the angle and adjacent side length are known, then the opposite side (the height of the tree) can be solved for with the tangent function. For example, if standing $100$ feet from the base of the tree the tip makes a 15 degree angle the height is given by:

theta = 15 * 180/pi
adjacent = 100
opposite = adjacent * tan(theta)
-466.6470644343919

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 or fist. How, the fist takes up about $10$ degree 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:

tan(10 * pi/180)
0.17632698070846498

Which for sake of memory we will say is $1/6$ (a $5$ percent error). So that answer is roughly $15$ feet:

30 * 3 / 6
15.0

Similarly, you can use your thumb instead of your first. To use your first you can multiply by $1/6$ the adjacent side, to use your thumb about $1/30$ as this approximates the tangent of $2$ degrees:

1/30, tan(2*pi/180)
(0.03333333333333333, 0.03492076949174773)

This could be reversed. If you know the height of something a distance away that is covered by your thumb or fist, then you would multiply that height by the appropriate amount to find your distance.

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:

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.

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 $\dots,-2,-1, 0, 1, 2, \dots$.

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:

plot(sin, 0, 4pi)
plot!(cos, 0, 4pi)

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 is also periodic, but not with period $2\pi$, but rather just $\pi$. A graph will show this. Here we avoid the vertical asymptotes by keeping them out of the plot domain and layering several plots.

Functions using degrees

Trigonometric function are functions of angles which have two common descriptions: in terms of degrees or radians. Degrees are common when right triangles are considered, radians much more common in general, as the relationship with arc-length holds in that $r\theta = l$, where $r$ is the radius of a circle and $l$ the length of the arc formed by angle $\theta$.

The two are related, as a circle of $2\pi$ radians and 360 degrees. So to convert from degrees into radians it takes multiplying by $2\pi/360$ and to convert from radians to degrees it takes multiplying by $360/(2\pi)$. The deg2rad and rad2deg functions are available for this task.

In Julia, the functions sind, cosd, tand, cscd, secd, and cotd are available to simplify the task of composing the two operations (that is sin(deg2rad(x)) is the same as sind(x)).

The sum-and-difference formulas

Consider the point on the unit circle $(x,y) = (\cos(\theta), \sin(\theta))$. In terms of $(x,y)$ (or $\theta$) is there a way to represent the angle found by rotating an additional $\theta$, that is what is $(\cos(2\theta), \sin(2\theta))$?

More generally, suppose we have two angles $\alpha$ and $\beta$, can we represent the values of $(\cos(\alpha + \beta), \sin(\alpha + \beta))$ using the values just involving $\beta$ and $\alpha$ separately?

According to Wikipedia the following figure (from mathalino.com) has ideas that date to Ptolemy:

Geometric picture

To read this, there are three triangles: the bigger (green with pink part) has hypotenuse $1$ (and adjacent and opposite sides that form the hypotenuses of the other two); the next biggest (yellow) hypotenuse $\cos(\beta)$, adjacent side (of angle $\alpha$) $\cos(\beta)\cdot \cos(\alpha)$, and opposite side $\cos(\beta)\cdot\sin(\alpha)$; and the smallest (pink) hypotenuse $\sin(\beta)$, adjacent side (of angle $\alpha$) $\sin(\beta)\cdot \cos(\alpha)$, and opposite side $\sin(\beta)\sin(\alpha)$.

This figure shows the following sum formula for sine and cosine:

\[ ~ \begin{align} \sin(\alpha + \beta) &= \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha)\\ \cos(\alpha + \beta) &= \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta). \end{align} ~ \]

Using the fact that $\sin$ is an odd function and $\cos$ an even function, related formulas for the difference $\alpha - \beta$ can be derived.

Taking $\alpha = \beta$ we immediately get the "double-angle" formulas:

\[ ~ \begin{align} \sin(2\alpha) &= 2\sin(\alpha)\cos(\alpha)\\ \cos(2\alpha) &= \cos(\alpha)^2 - \sin(\alpha)^2. \end{align} ~ \]

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 $\cos(2\alpha) = 2\cos(\alpha)^2 - 1$ or $1 - 2\sin(\alpha)^2$.

Applying the above with $\alpha = \beta/2$, we get that $\cos(\beta) = 2\cos(\beta/2)^2 -1$, which rearranged yields the "half-angle" formula: $\cos(\beta/2)^2 = (1 + \cos(\beta))/2$.

Example

Consider the expressions $\cos((n+1)\theta)$ and $\cos((n-1)\theta)$. These can be re-expressed as:

\[ ~ \begin{align} \cos((n+1)\theta) &= \cos(n\theta + \theta) = \cos(n\theta) \cos(\theta) - \sin(n\theta)\sin(\theta), \text{ and}\\ \cos((n-1)\theta) &= \cos(n\theta - \theta) = \cos(n\theta) \cos(-\theta) - \sin(n\theta)\sin(-\theta). \end{align} ~ \]

But $\cos(-\theta) = \cos(\theta)$, whereas $\sin(-\theta) = -\sin(\theta)$. Using this, we add the two formulas above to get:

\[ ~ \cos((n+1)\theta) = 2\cos(n\theta) \cos(\theta) - \cos((n-1)\theta). ~ \]

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 $\cos(\theta)$.

Inverse trigonometric functions

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 example, the arcsine function is defined for $-1 \leq x \leq 1$ and has a range of $-\pi/2$ to $\pi/2$:

plot(asin, -1, 1)

The arctangent has domain of all real $x$. It has shape given by:

plot(atan, -10, 10)

The horizontal asymptotes are $y=\pi/2$ and $y=-\pi/2$.

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.

However, $\arcsin(\sin(x))$ is defined for all $x$, but only equals $x$ when $x$ is in $[-\pi/2, \pi/2]$. The output, or range, of the $\arcsin$ function is restricted to that interval.

This can be limiting at times. A common case is to find the angle in $[0, 2\pi)$ corresponding to a point $(x,y)$. In the simplest case (the first and fourth quadrants) this is just given by $\arctan(y/x)$. But with some work, the correct angle can be found for any pair $(x,y)$. As this is a common desire, the atan function with two arguments, atan(y,x), is available. This function returns a value in $(-\pi, \pi]$.

For example, this will not give back $\theta$ without more work to identify the quadrant:

theta = 3pi/4                     # 2.35619...
x,y = (cos(theta), sin(theta))    # -0.7071..., 0.7071...
atan(y/x)
-0.7853981633974484

But,

atan(y, x)
2.356194490192345
Example

A (white) light shining through a prism will be deflected depending on the material of the prism and the angles involved (cf. the link for a figure). The relationship can be analyzed by tracing a ray through the figure and utilizing Snell's law. If the prism has index of refraction $n$ then the ray will deflect by an amount $\delta$ that depends on the angle, $\alpha$ of the prism and the initial angle ($\theta_0$) according to:

\[ ~ \delta = \theta_0 - \alpha + \arcsin(n \sin(\alpha - \arcsin(\frac{1}{n}\sin(\theta_0)))). ~ \]

If $n=1.5$ (glass), $\alpha = \pi/3$ and $\theta_0=\pi/6$, find the deflection (in radians).

We have:

n, alpha, theta0 = 1.5, pi/3, pi/6
delta = theta0 - alpha + asin(n * sin(alpha - asin(sin(theta0)/n)))
0.8219769749498015

For small $\theta_0$ and $\alpha$ the deviation is approximated by $(n-1)\alpha$. Compare this approximation to the actual value when $\theta_0 = \pi/10$ and $\alpha=\pi/15$.

We have:

n, alpha, theta0 = 1.5, pi/15, pi/10
delta = theta0 - alpha + asin(n * sin(alpha - asin(sin(theta0)/n)))
delta, (n-1)*alpha
(0.10763338241545499, 0.10471975511965977)

The approximation error is about 2.7 percent.

Example

The AMS has an interesting column on rainbows the start of which uses some formulas from the previous example. Click through to see a ray of light passing through a spherical drop of water, as analyzed by Descartes. The deflection of the ray occurs when the incident light hits the drop of water, then there is an internal deflection of the light, and finally when the light leaves, there is another deflection. The total deflection (in radians) is $D = (i-r) + (\pi - 2r) + (i-r) = \pi - 2i - 4r$. However, the incident angle $i$ and the refracted angle $r$ are related by Snell's law: $\sin(i) = n \sin(r)$. The value $n$ is the index of refraction and is $4/3$ for water. (It was $3/2$ for glass in the previous example.) This gives

\[ ~ D = \pi + 2i - 4 \arcsin(\frac{1}{n} \sin(i)). ~ \]

Graphing this for incident angles between $0$ and $\pi/2$ we have:

n = 4/3
D(i) = pi + 2i - 4 * asin(sin(i)/n)
plot(D, 0, pi/2)

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:

rad2deg(1.0)
57.29577951308232

(Using calculus it can be seen to be $\arccos(((n^2-1)/3)^{1/2})$.)

Example: The Chebyshev Polynomials

Consider again this equation derived with the sum-and-difference formula:

\[ ~ \cos((n+1)\theta) = 2\cos(n\theta) \cos(\theta) - \cos((n-1)\theta). ~ \]

Let $T_n(x) = \cos(n \arccos(x))$. Calling $\theta = \arccos(x)$ for $-1 \leq x \leq x$ we get a relation between these functions:

\[ ~ T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x). ~ \]

We can simplify a few: For example, when $n=0$ we see immediately that $T_0(x) = 1$, the constant function. Whereas with $n=1$ we get $T_1(x) = \cos(\arccos(x)) = x$. Things get more interesting as we get bigger $n$, for example using the equation above we get $T_2(x) = 2xT_1(x) - T_0(x) = 2x\cdot x - 1 = 2x^2 - 1$. Continuing, we'd get $T_3(x) = 2 x T_2(x) - T_1(x) = 2x(2x^2 - 1) - x = 4x^3 -3x$.

A few things become clear from the above two representations:

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 can 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):

T4(x) = (8x^4 - 8x^2 + 1) / 8
q(x) = (x+3/5)*(x+1/5)*(x-1/5)*(x-3/5)
plot(abs  T4, -1,1)
plot!(abs  q, -1,1)

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 (with $x^2 - y^2 = 1$. We define the hyperbolic sin ($\sinh$) and hyperbolic cosine ($\cosh$) through $(\cosh(\theta), \sinh(\theta)) = (x,y)$.

These can be expressed using the exponential function as:

\[ ~ \begin{align} \sinh(x) &= \frac{e^x - e^{-x}}{2}\\ \cosh(x) &= \frac{e^x + e^{-x}}{2}. \end{align} ~ \]

The hyperbolic tangent is then the ratio of $\sinh$ and $\cosh$. As well, three inverse hyperbolic functions can be defined.

Questions

Question

What is bigger $\sin(1.23456)$ or $\cos(6.54321)$?

Question

Let $x=\pi/4$. What is bigger $\cos(x)$ or $x$?

Question

The cosine function is a simple tranformation of the sine function. Which one?

Question

Graph the secant function. The vertical asymptotes are at?

Question

A formula due to Bhaskara I dates to around 650AD and gives a rational function approximation to the sine function. In degrees, we have

\[ ~ \sin(x^\circ) \approx \frac{4x(180-x)}{40500 - x(180-x)}, \quad 0 \leq x \leq 180. ~ \]

Plot both functions over $[0, 180]$. What is the maximum difference between the two to two decimal points? (You may need to plot the difference of the functions to read off an approximate answer.)

Question

Solve the following equation for a value of $x$ using acos:

\[ ~ \cos(x/3) = 1/3. ~ \]

Question

For any postive integer $n$ the equation $\cos(x) - nx = 0$ has a solution in $[0, \pi/2]$. Graphically estimate the value when $n=10$.

Question

The sine function is an odd function.

Question

The hyperbolic sine satisfies this formula:

\[ ~ \sinh(\theta + \beta) = \sinh(\theta)\cosh(\beta) + \sinh(\beta)\cosh(\theta). ~ \]

Is this identical to the pattern for the regular sine function?

The hyperbolic cosine satisfies this formula:

\[ ~ \cosh(\theta + \beta) = \cosh(\theta)\cosh(\beta) + \sinh(\beta)\sinh(\theta). ~ \]

Is this identical to the pattern for the regular sine function?