Trigonometric Functions

Trigonometric functions are real-valued functions that relate the measure of an angle in a right triangle to the ratios of two of its sides. They play a fundamental role in geometry and are widely used in many scientific fields, such as navigation, mechanics, astronomy, geodesy, and more. Trigonometric functions are also among the simplest examples of periodic functions, making them essential for modeling periodic phenomena (such as waves) and for applications like Fourier analysis.

A function \(f\) is periodic if it repeats itself in regular intervals.

A function \(f\) is periodic if there exists a constant \(P>0\) such that$$f(x+P) = f(x), \forall x$$

• The period is the length of one full repetition or cycle. It is the smallest positive value \(p\) such that \(f(x+p) = f(x)\) for all \(x\).
• The principal axis or mean line is the horizontal line about which the wave oscillates, given by $$y = \frac{\text{maximum value} + \text{minimum value}}{2}$$
• The amplitude is the distance from the principal axis to a maximum or minimum point, given by $$\frac{\text{maximum value} - \text{minimum value}}{2}$$

Among the many types of periodic functions, this chapter will focus on those that exhibit a smooth, wave-like pattern. To model these sinusoidal functions, we will use the sine and cosine trigonometric functions.

Let \(M(\cos x, \sin x)\) be the point on the unit circle corresponding to an angle \(x\) (in radians).

• The angle \(x\) on the unit circle corresponds to the input for the sine function.
• The \(y\)-coordinate of point \(M\) on the unit circle, \(\sin x\), gives the output of the sine function.

Thus, plotting \(x \mapsto \sin x\) produces the graph of the sine function.
See for example: \href{https://www.geogebra.org/m/j7w29vj4}{GeoGebra demo}.

The sine function, denoted \(\sin\), is defined by \(x\mapsto \sin(x)\), where \(x\) is interpreted as an angle in radians.

Complete the following table with the values of the sine function at key angles:

\(x\)	0	\(\dfrac{\pi}{6}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{3}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{2 \pi}{3}\)	\(\dfrac{3 \pi}{4}\)	\(\dfrac{5 \pi}{6}\)	\(\pi\)
\(\sin (x)\)

If we instead project the values of \(\cos x\) from the unit circle onto a graph, we obtain the graph of the cosine function \(x \mapsto \cos x\).

The cosine function, denoted \(\cos\), is defined by \(x\mapsto\cos(x)\), where \(x\) is interpreted as an angle in radians.

Complete the following table with the values of the cosine function at key angles:

\(x\)	0	\(\dfrac{\pi}{6}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{3}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{2 \pi}{3}\)	\(\dfrac{3 \pi}{4}\)	\(\dfrac{5 \pi}{6}\)	\(\pi\)
\(\cos (x)\)

• Periodicity: Both functions are periodic with a period of \(2\pi\). \(\sin(x+2\pi) = \sin(x)\) and \(\cos(x+2\pi) = \cos(x)\).
• Domain and Range: The domain is \(\mathbb{R}\). The range is \([-1, 1]\).
• Symmetry: Cosine is an even function (\(\cos(-x)=\cos(x)\)). Sine is an odd function (\(\sin(-x)=-\sin(x)\)).
• Amplitude: The amplitude of the base functions is 1.

Now that we are familiar with the graphs of \(y = \sin x\) and \(y = \cos x\), we can use transformations to graph more complicated trigonometric functions.

For functions of the form \(y = a\sin(b(x-c)) + d\) and \(y = a\cos(b(x-c)) + d\):

• \(|a|\) is the amplitude.
• \(\frac{2\pi}{|b|}\) is the period.
• \(c\) is the phase shift (horizontal translation).
• \(d\) is the vertical shift (the principal axis is \(y=d\)).

For real parameters \(a,b,c,d\) with \(b\neq 0\), the graph of \(y = a\sin\!\big(b(x-c)\big) + d\) is obtained from the graph of \(y=\sin x\) by:

1. a vertical stretch by factor \(|a|\); if \(a<0\), there is also a reflection across the horizontal axis,
2. a horizontal stretch by factor \(\dfrac{1}{|b|}\); if \(b<0\), there is also a reflection across the vertical axis,
3. a horizontal translation by \(c\) units (to the right if \(c>0\), left if \(c<0\)) and a vertical translation by \(d\) units (up if \(d>0\), down if \(d<0\)).

Sketch the graph of \(y=\sqrt{2} \sin x-1\) for \(0 \leqslant x \leqslant 2 \pi\).

The tangent function is defined by the ratio:$$\textcolor{colordef}{\tan(x) = \frac{\sin x}{\cos x}}$$

The graph of \(y = \tan(x) = \frac{\sin x}{\cos x}\) has:

• Domain: \(\{x \in \mathbb{R} \mid x \neq \frac{\pi}{2} + k\pi,\forall k\in\mathbb{Z}\}\) (all real numbers \(x\) except those for which \(\cos x = 0\))
• Period: \(\pi\).
• Vertical Asymptotes: at \(x = \frac{\pi}{2} + k\pi\) for any integer \(k\).
• Range: \(\mathbb{R}\).

• The cosecant is \(\csc(x) = \frac{1}{\sin(x)}\), for \(\sin(x) \neq 0\).
• The secant is \(\sec(x) = \frac{1}{\cos(x)}\), for \(\cos(x) \neq 0\).
• The cotangent is \(\cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}\), for \(\sin(x) \neq 0\).

The graph of \(y = \csc(x) = \frac{1}{\sin(x)}\) is shown below.

• Period: \(2\pi\).
• Vertical Asymptotes: At \(x = k\pi\) (where \(\sin x = 0\)).
• Range: \(({-\infty}, -1] \cup [1, \infty)\).

The graph of \(y = \sec(x) = \frac{1}{\cos(x)}\) is shown below.

• Period: \(2\pi\).
• Vertical Asymptotes: At \(x = \frac{\pi}{2} + k\pi\) (where \(\cos x = 0\)).
• Range: \(({-\infty}, -1] \cup [1, \infty)\).

The graph of \(y = \cot(x) = \frac{1}{\tan(x)}\) is shown below.

• Period: \(\pi\).
• Vertical Asymptotes: At \(x = k\pi\) (where \(\sin x = 0\)).
• Range: \(\mathbb{R}\).

The inverse trigonometric functions, also known as arc functions, allow us to find the angle given the value of a trigonometric ratio. They are the inverses of the sine, cosine, and tangent functions, but with restricted domains to make them one-to-one.

The inverse sine function, denoted \(\arcsin\) or \(\sin^{-1}\), is defined as the function that returns the angle \(x\) such that \(\sin x = y\), where \(-1 \le y \le 1\) and \(-\frac{\pi}{2} \le x \le \frac{\pi}{2}\).

• Domain: \([-1, 1]\)
• Range: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

The graph of \(\textcolor{colordef}{y = \arcsin x}\) is the reflection of \(\textcolor{colorprop}{y = \sin x}\) over the line \(\textcolor{olive}{y = x}\), restricted to the principal branch.

The inverse cosine function, denoted \(\arccos\) or \(\cos^{-1}\), is defined as the function that returns the angle \(x\) such that \(\cos x = y\), where \(-1 \le y \le 1\) and \(0 \le x \le \pi\).

• Domain: \([-1, 1]\)
• Range: \([0, \pi]\)

The graph of \(\textcolor{colordef}{y = \arccos x}\) is the reflection of \(\textcolor{colorprop}{y = \cos x}\) over the line \(\textcolor{olive}{y = x}\), restricted to the principal branch.

The inverse tangent function, denoted \(\arctan\) or \(\tan^{-1}\), is defined as the function that returns the angle \(x\) such that \(\tan x = y\), where \(y \in \mathbb{R}\) and \(-\frac{\pi}{2} < x < \frac{\pi}{2}\).

• Domain: \(\mathbb{R}\)
• Range: \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

The graph of \(y = \arctan x\) is the reflection of \(y = \tan x\) over the line \(y = x\), restricted to the principal branch.

• \(\arcsin(\sin x) = x\) for \(x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
• \(\arccos(\cos x) = x\) for \(x \in [0, \pi]\)
• \(\arctan(\tan x) = x\) for \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Evaluate \(\arcsin\left(\frac{\sqrt{3}}{2}\right)\).

1. Use identities to simplify the equation into a form involving a single trigonometric function, e.g., \(\sin(X)=k\).
2. Find the principal value (the first solution) using an inverse function, \(X = \arcsin(k)\).
3. Use the symmetry and periodicity of the function to find all solutions within one period.
4. Add multiples of the period (\(2k\pi\) or \(k\pi\)) to find the general solution or all solutions in a specified domain.

Find the general solution to the equation \(2\sin(3x) = \sqrt{3}\), and hence find all solutions in the interval \(0 \le x \le \pi\).

To find a sinusoidal model of the form \(y = a\sin(b(x-c)) + d\) that fits a set of periodic data:

1. Find the vertical shift, \(d\): Calculate the principal axis, which is the average of the maximum and minimum values. $$d = \frac{\text{max value} + \text{min value}}{2}$$
2. Find the amplitude, \(a\): Calculate half the distance between the maximum and minimum values. $$a = \frac{\text{max value} - \text{min value}}{2}$$
3. Find the period and the parameter \(b\): Determine the period, \(P\), which is the length of one full cycle of the data. Use the formula \(P = \frac{2\pi}{b}\) to solve for \(b\). $$b = \frac{2\pi}{P}$$
4. Find the phase shift, \(c\): Substitute a data point \((x,y)\) into the equation \(y = a\sin(b(x-c))+d\) and solve for the horizontal shift \(c\). It is often easiest to choose a point where the function crosses its principal axis, as this corresponds to a point where the sine function is zero.

The mean monthly maximum temperatures for Cape Town, South Africa are shown below:

Month (\(t\))	1	2	3	4	5	6	7	8	9	10	11	12
Temp (\(T\))	28	27	25.5	22	18.5	16	15	16	18	21.5	24	26

Find a sinusoidal function of the form \(T = a\sin(b(t-c))+d\) to model the temperature.