\chapter{Trigonometry}
\section{Angles and their measure}
The word \textit{Trigonometry} can be broken down into its Greek origins: \textit{trigonon} (triangle) and
\textit{metron} (measure). So trigonometry studies "measurement of triangles". And \textit{triangle}
means a figure with three angles. So we begin by discussing what an angle is.
\subsection{Angles}
An angle is determined by two straight line segments that meet at a common point, called the
\textit{vertex}. We distinguish between an \textit{initial side} and a \textit{terminal side}
for the angle, and we think of the angle as being generated when the initial side rotates
and reaches the terminal side. Just like in algebra \(x\) is the most common symbol used, the most common symbol for angles
is the Greek letter \(\theta\) (theta). If we need a second letter we often use the Greek \(\phi\) (just like in algebra we would use \(y\) when we need a second symbol).
\[
\img{U4_1F1aa.png}{}{25em}{}
\]
An angle
Angles can be positive or negative, depending on the direction of rotation. We take a
counter-clockwise rotation (against the clock) to be positive, and a clockwise rotation
to be negative. So the angle in the picture above is positive, while the angle in the following picture
is negative.
\[
\img{U4_1F2.png}{}{20em}{}
\]
A negative angle
We say that an angle is in \textit{standard position} if its initial side is along the
positive \(x\)-axis. So the angle in the previous picture is in standard position.
The angle in the next picture is not standard position.
\[
\img{U4_1F3.png}{}{16em}{}
\]
A positive angle, not in standard position
A basic difference between moving along a straight line and rotating around a circle is that
in the second case we come back to the same position after a full rotation. Or, we
can reach the same position by rotating in the opposite direction.
\subsection{Co-terminal angles}
Taking the idea of the rotation to define angles further, we can also consider angles that are obtained by rotating
beyond a full rotation, as in
the picture below:
\[
\img{U4_1F1a.png}{}{25em}{}
\]
The angle \(\phi\) in this picture is different from the angle \(\theta\) in Figure 1 above,
even though they have the
same initial and terminal sides. From a given initial position, we can reach the same terminal position
in more than one way. So angles may have the same initial and
terminal sides and still be different angles.
When two angles have the same initial and terminal sides, we say that the two angles are \textit{co-terminal}.
Notice that according to this definition, an angle is always co-terminal with itself.
Angles whose terminal side is on the \(x\) or \(y\) axis are called \textit{quadrantal angles}.
The following picture shows two quadrantal angles. \(\theta_1\) is positive, and \(\theta_2\) is
negative.
\[
\img{U4_1F.png}{}{30em}{}
\]
Two quadrantal angles
If an angle is in standard position, and it is not a quadrantal angle,
the \textit{quadrant of the angle} is the quadrant that contains the terminal side.
The following pictures
show two co-terminal angles, one positive and one negative, for each of the four quadrants.
\[\begin{array}{cc}
\img{U4_1F4a.png}{}{16em}{} & \img{U4_1F5b.png}{}{16em}{} \\
\theta_1 \mbox{ is positive}, \theta_2 \mbox{ is negative } & \theta_3 \mbox{ is positive}, \theta_4 \mbox{ is negative }\\[2ex]
\img{U4_1F6c.png}{}{16em}{}& \img{U4_1F7d.png}{}{16em}{}\\
\theta_5 \mbox{ is positive}, \theta_6 \mbox{ is negative } &\theta_7 \mbox{ is positive}, \theta_8 \mbox{ is negative }
\end{array}\]
The next picture shows three co-terminal angles: \(\theta_1\) and \(\theta_3\) are positive, and
\(\theta_2\) is negative.
\[\img{U4_1F8.png}{}{25em}{}\\
\]
Three co-terminal angles
\subsection{Degree measure of angles}
There are two main ways to measure the size of an angle: degrees and radians.
For degrees, we take a full circle to have a measure of 360 degrees:
\[
\img{U4_1F10.png}{}{16em}{}
\]
\(360^\circ = 1\mbox{ full revolution around a circle}\)
To indicate that we are measuring angles in degrees, we must always use the degree symbol \(^\circ\).
If we divide a full revolution by \(4\) we get a \textit{right angle}, with degree measure
\(360/4 = 90^\circ \):
\[
\begin{array}{cc} \img{U4_1F11.png}{}{16em}{} &
\img{U4_1F12.png}{}{16em}{}\\
\mbox{a positive right angle} & \mbox{a negative right angle}
\end{array}
\]
Two right angles make a \textit{straight angle}, of measure \(180^\circ\):
\[
\img{U4_1F13.png}{}{16em}{}
\]
A straight angle
Three right angles make a \(270^\circ\) angle:
\[
\img{U4_1F14.png}{}{16em}{}
\]
If we divide a right angle into \(2\) we get \(45^\circ\) angles:
\[
\img{U4_1F15a.png}{}{12em}{}
\]
If we divide a right angle into \(3\) we get \(30^\circ\) angles:
\[
\img{U4_1F15.png}{}{12em}{}
\]
To find co-terminal angles given in degrees, we can add or subtract \(360^\circ\). Adding
\(360^\circ\) means that starting from the terminal side, we go around the circle once in
counter-clockwise direction. Subtracting \(360^\circ\) means going around in clockwise
direction. And we can keep adding and subtracting more than once to find more and more
co-terminal angles.
\begin{example} \
\begin{itemize}
\item
Suppose we are given \(\theta= 120^\circ\).
A positive co-terminal angle is found by adding \(360^\circ\):
\[\theta_1=120+360=480^\circ.\]
A negative co-terminal angle is found by subtracting \(360^\circ\):
\[\theta_2=120-360=-240^\circ.\]
\item
If we start with \(\theta = 495^\circ\) and we want to find a negative co-terminal angle,
subtracting \(360^\circ\) once we find \(495-360=135^\circ\), that is still positive,
so we subtract \(360^\circ\) again and find that \(135-360=-225^\circ\) is a negative angle
co-terminal with \(495^\circ\).
\end{itemize}
\end{example}
| Finding co-terminal angles in degrees |
|---|
| \begin{itemize}
\item
Given an angle \(\theta\) in degrees, to find a second, positive angle co-terminal with \(\theta\), add \(360^\circ\) one or more times,
until you find a positive angle.
\item
Given an angle \(\theta\) in degrees, to find a second, negative angle co-terminal with \(\theta\), subtract \(360^\circ\) one or more times,
until you find a negative angle.
\end{itemize} |
\subsection{Radian measure of angles}
The choice of \(360\) for the number of degrees in a full revolution is just a convenient one, because
\(360\) is a number that can be evenly divided by many numbers. The \textit{radian measure} uses
a different approach: the radian measure of a full revolution is the length of the circumference
in a circle of radius \(1\). Since the formula for the circumference is
\(C=2\pi r\), using \(r=1\) this length is \(2\pi(1)=2\pi\).
So we find \[ 1\mbox{ full revolution }= 360^\circ = 2\pi \mbox{ radians }.\]
This means that to get a straight angle we just take half the length:
\[180^\circ = \pi \mbox{ radians},\]
and to get a right angle we divide by \(2\) again:
\[90^\circ = \frac{\pi}{2} \mbox{ radians}.\]
If we divide the equation \(180^\circ = \pi \mbox{ radians}\) by \(\pi\), we find
\[1\mbox{ radian} =\frac{180}{\pi}\approx 57.3^\circ,\]
and if we divide the same equation by \(180\), we find
\[1^\circ = \frac{\pi}{180} \approx 0.01745\mbox{ radians}.\]
This means that to convert radian measure to degrees, we need to multiply by \(180/\pi\), and
to convert degrees to radians we need to multiply by \(\pi/180\).
Note that \(1\) radian is much larger than \(1^\circ\) (it's about \(57^\circ\)). So when we measure
angles in degrees, the number is larger than when we measure it in radians. You can use this
fact to remember which factor to use: if we want to convert from radians to degree, then the
number must become larger, so we need to keep the \(180\) at the top of the fraction and the
right factor is \(180/\pi\). But if we want to convert from degrees to radians, the number must
become smaller, so we want the \(180\) in the denominator, and the right factor is \(\pi/180\).
| Converting between degrees and radians |
|---|
| \begin{itemize}
\item
To change from radians to degrees: multiply by \(\displaystyle \frac{180}{\pi}\)
\item
To change from degrees to radians: multiply by \(\displaystyle \frac{\pi}{180}\)
\end{itemize}
|
It is important to remember that the measure of an angle given by just a number is always
understood to be in radians, unless there is the symbol \(^\circ\) for degrees. Most of the time
we will not write the word "radians". So for example
\(\theta =90\) means that the measure of \(\theta\) is \(90\) radians. If we want to say it is
\(90\) degrees, we must write \(\theta = 90^\circ\).
This is true whether or not the number includes \(\pi\). What makes an angle measure radians is
not the presence of \(\pi\), but it is the absence of the degree symbol \(^\circ\). So \(\theta=1\) means \(1\) radian, \(\theta=1^\circ\) means \(1\) degree, \(\theta = \pi\) means \(\pi\) radians, and
\(\theta = \pi^\circ\) means \(\pi\) degrees.
\begin{example} \
\begin{itemize}
\item
We find the radian measure of \(30^\circ\), \(45^\circ\) and \(60^\circ\).
\[
\begin{array}{rcl}30^\circ &=&\displaystyle\frac{\pi}{180}\cdot 30=\frac{\pi}{6}\\[2ex]
45^\circ &=&\displaystyle\frac{\pi}{180}\cdot 45=\frac{\pi}{4}\\[2ex]
60^\circ &=&\displaystyle\frac{\pi}{180}\cdot 60=\frac{\pi}{3}
\end{array}
\]
\item
Now we find the degree measure of a few angles given in radians:
\[
\begin{array}{rcl} 2 &=&\displaystyle\frac{180}{\pi}\cdot 2=\left(\frac{360}{\pi}\right)^\circ\approx
114.59^\circ\\[2ex]
\displaystyle \frac{3\pi}{4} &=&\displaystyle\frac{180}{\pi}\cdot \frac{3\pi}{4}=135^\circ\\[2ex]
\displaystyle \frac{5\pi}{3} &=&\displaystyle\frac{180}{\pi}\cdot\frac{5\pi}{3}=300^\circ
\end{array}
\]
\end{itemize}
\end{example}
\begin{example}
If we want to find co-terminal angles in radians, we need to add or subtract \(2\pi\), because
\(2\pi=360^\circ\). So a positive angle co-terminal with \(\displaystyle \frac{5\pi}{4}\) is
\[\frac{5\pi}{4}+2\pi=\frac{5\pi}{4}+\frac{8\pi}{4}=\frac{13\pi}{4},\]
and a negative angle co-terminal with \(\displaystyle \frac{4\pi}{3} \) is
\[\frac{4\pi}{3} - 2\pi= \frac{4\pi}{3} - \frac{6\pi}{3} = -\frac{2\pi}{3}.\]
\end{example}
| Finding co-terminal angles in radians |
|---|
| \begin{itemize}
\item
Given an angle \(\theta\) in radians, to find a second, positive angle co-terminal with \(\theta\),
add \(2\pi\) one or more times,
until you find a positive angle.
\item
Given an angle \(\theta\) in radians, to find a second, negative angle co-terminal
with \(\theta\), subtract \(2\pi\) one or more times,
until you find a negative angle.
\end{itemize} |
Problems
\problem
Let \(\theta=135^\circ\).
\begin{enumerate}
\item
Find a different, positive angle that is co-terminal with \(\theta\).
\item
Find a negative angle that is co-terminal with \(\theta\).
\end{enumerate}
\begin{sol}
\begin{enumerate}
\item
\(\theta_1=135+360=495^\circ\) is a positive angle co-terminal with \(\theta\).
\item
\(\theta_2= 135 - 360 = -225^\circ\) is a negative angle co-terminal with \(\theta\).
\end{enumerate}
\end{sol} \mproblem
Let \(\theta=210^\circ\).
\begin{enumerate}
\item
Find a different, positive angle that is co-terminal with \(\theta\).
\item
Find a negative angle that is co-terminal with \(\theta\).
\end{enumerate}
\problem
Let \(\theta=690^\circ\).
\begin{enumerate}
\item
Find a different, positive angle \(\theta_1\) co-terminal with \(\theta\), in degrees.
\item
Find a negative angle \(\theta_2\) co-terminal with \(\theta\), in degrees.
\item
Convert \(\theta\), \(\theta_1\) and \(\theta_2\) to radians.
\end{enumerate}
\begin{sol}
\begin{enumerate}
\item We could add \(360\) and get a larger co-terminal angle. But since \(690\) is greater
than \(360\), we can also subtract \(360\) and get a smaller positive co-terminal angle:
\(\theta_1=690 -360=330^\circ\).
\item
Subtracting \(360\) again, we find \(\theta_2=330-360=-30^\circ\).
\item
\[\theta=690\cdot\frac{\pi}{180}=\frac{23\pi}{6}.\]
\[\theta_1=330\cdot \frac{\pi}{180}=\frac{11\pi}{6}.\]
\[\theta_2=-30\cdot\frac{\pi}{180}=-\frac{\pi}{6}.\]
\end{enumerate}
\end{sol} \mproblem
Let \(\theta=585^\circ\).
\begin{enumerate}
\item
Find a different, positive angle \(\theta_1\) co-terminal with \(\theta\), in degrees.
\item
Find a negative angle \(\theta_2\) co-terminal with \(\theta\), in degrees.
\item
Convert \(\theta\), \(\theta_1\) and \(\theta_2\) to radians.
\end{enumerate}