\chapter{Trigonometry}
\section{Applications of trig functions}
In Unit 4.3 we found all the trig functions for \(30^\circ\), \(45^\circ\) and \(60^\circ\).
Then in Unit 4.4 we found the trig functions for quadrantal angles, and in
Unit 4.5 the trig values
for all the multiples of \(30^\circ\), \(45^\circ\) and \(60^\circ\) in all quadrants.
All the values found so far were exact. For example, \(\sin (60^\circ) = \sqrt{3}/2\) is an
exact value. Knowing the exact value is often preferable
to only knowing a decimal approximation such as \(\sin (60^\circ)=0.866\ldots\), that
is given by a scientific calculator.
But it is difficult to find exact values for angles other than the ones we have already seen.
So in this unit we will practice using a calculator to find decimal approximations of
the trig functions of angles for which an exact value is not available.
\subsection{Decimal approximations of trig functions}
All scientific calculators have \(\sin\), \(\cos\) and \(\tan\) keys, and will
evaluate angles both in degrees and in radians.
An important thing to remember when using a calculator for trigonometry is to make sure
that it is set in the right mode for the calculation (either degrees, or radians).
\begin{example} \
\begin{itemize}
\item
To find \(\sin(42^\circ)\), make sure the calculator is in degree mode. Then press the keys
\[\fbox{\(\sin\)}( \hspace{1ex} \fbox{\(4\)} \hspace{1ex}\fbox{\(2\)} \hspace{1ex} \fbox{)} \hspace{1ex}
\fbox{=}\]
and find
\[\sin(42^\circ) \approx 0.669.\]
\item Suppose now we want \(\sin(42)\). This is an angle in radians, so we need to change the
calculator mode to radians, and we find
\[\fbox{\(\sin\)}( \hspace{1ex} \fbox{\(4\)} \hspace{1ex}\fbox{\(2\)} \hspace{1ex} \fbox{)} \hspace{1ex}
\fbox{=} \hspace{1ex} -0.916\ldots\]
\item To find \(\cos(95^\circ)\), put the calculator in degree mode, and find
\[\fbox{\(\cos\)}( \hspace{1ex} \fbox{\(9\)} \hspace{1ex}\fbox{\(5\)} \hspace{1ex} \fbox{)} \hspace{1ex}
\fbox{=}\hspace{1ex} -0.087\ldots \]
\item For \(\cos(95)\), we need radian mode:
\[\fbox{\(\cos\)}( \hspace{1ex} \fbox{\(9\)} \hspace{1ex}\fbox{\(5\)} \hspace{1ex} \fbox{)} \hspace{1ex}
\fbox{=}\hspace{1ex} 0.730\ldots \]
\item To find \(\displaystyle \tan\left(\frac{2\pi}{5}\right)\), use radian mode:
\[\fbox{\(\tan\)}( \hspace{1ex} \fbox{\(2\)} \hspace{1ex}\fbox{\(\pi\)} \hspace{1ex} \fbox{\(\div\)} \hspace{1ex}
\fbox{5}\hspace{1ex}\fbox{ )} \hspace{1ex} \fbox{\(=\)}\hspace{1ex} 3.077\ldots \]
\end{itemize}
\end{example}
To find the other three functions
\(\csc\), \(\sec\), or \(\cot\) we need to use the reciprocal key \(\fbox{\(x^{-1}\)}\)
after evaluating \(\sin\), \(\cos\), or \(\tan\). Do not use the \(\fbox{\(\sin^{-1}\)}\), \(\fbox{\(\cos^{-1}\)}\), \(\fbox{\(\tan^{-1}\)}\)
keys for this purpose: those keys are for the inverse trig functions that will be discussed later in
this chapter.
\begin{example} \
\begin{itemize}
\item To find \(\sec (12^\circ)\), first we find \(\cos(12^\circ)\), then we use the
\(\fbox{\(x^{-1}\)}\) key to find the reciprocal. In degree mode, we type:
\[\fbox{\(\cos\)}( \hspace{1ex} \fbox{\(1\)} \hspace{1ex}\fbox{\(2\)} \hspace{1ex} \fbox{)} \hspace{1ex}
\fbox{\(=\)}\hspace{1ex} \fbox{\(x^{-1}\)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} 1.022\ldots \]
\item
To find \(\displaystyle \cot\left(\frac{3\pi}{7}\right)\), first we use the \(\tan\) key, then the reciprocal key. In radian mode:
\[\fbox{\(\tan\)}( \hspace{1ex} \fbox{\(3\)} \hspace{1ex}\fbox{\(\pi\)} \hspace{1ex} \fbox{\(\div\)} \hspace{1ex}
\fbox{\(7\)}\hspace{1ex} \fbox{)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex}
\fbox{\(x^{-1}\)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} 0.228\ldots \]
\item
To find \(\csc(39^\circ)\) use (in degree mode)
\[\fbox{\(\sin\)}( \hspace{1ex} \fbox{\(3\)} \hspace{1ex}\fbox{\(9\)} \hspace{1ex} \fbox{)} \hspace{1ex}
\fbox{\(=\)}\hspace{1ex} \fbox{\(x^{-1}\)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} 1.589\ldots \]
\end{itemize}
\end{example}
Of course the calculator will not evaluate trig functions when they are not defined. For example, try
to find \(\tan(90^\circ)\) and see what you get.
\subsection{Solving triangles}
Using the Pythagorean theorem, we can find the third side of a right triangle as long as we know
two sides. With trigonometry, we can do something more: we can find all sides of a right triangle as soon
as we know just one side, and one of the acute angles.
\begin{example}
We are given the triangle in the picture:
\[\img{U4_6F1.png}{}{15em}{}\]
and we want to find all sides, rounded to three decimal places.
First label the sides:
\[\img{U4_6F2.png}{}{15em}{}\]
Then use the appropriate trig function: since we already know the \(\mbox{adj}\) side, we use the
cosine function:
\[\cos(24^\circ) =\frac{\mbox{adj}}{\mbox{hyp}}=\frac{15}{c}.\]
We can now solve this equation for \(c\), using a calculator at the last step:
\[\begin{array}{rcll}
\cos(24^\circ)& = & \displaystyle \frac{15}{c} & \\[2ex]
\color{red}{c} \cos(24^\circ)& = & \displaystyle \cancel{\color{red}{c}} \frac{15}{\cancel{c}} & \mbox{Multiply both sides by }c \\[2ex]
c \cos(24^\circ) & = & 15 & \mbox{Simplify}\\[2ex]
\displaystyle \frac{c \cancel{\cos(24^\circ)}}{\cancel{\color{red}{\cos(24^\circ)}}} & = & \displaystyle \frac{15}{\color{red}{\cos(24^\circ)}} & \mbox{Divide both sides by }\cos(24^\circ) \\[2ex]
c &= &\displaystyle \frac{15}{\cos(24^\circ)} & \mbox{Simplify, and get the exact answer}\\[2ex]
c & \approx & 16.419 & \mbox{Use a calculator to approximate the answer}
\end{array}
\]
To find \(a\), we could use the Pythagorean theorem. But we can also use the tangent function:
\[\tan(24^\circ) = \frac{\mbox{Opp}}{\mbox{Adj}} =\frac{a}{15},\]
and we solve the equation for \(a\):
\[\begin{array}{rcll}
\tan(24^\circ)& =& \displaystyle \frac{a}{15} & \\[2ex]
15 \tan(24^\circ)& = & \displaystyle a & \mbox{Multiply both sides by }15 \\[2ex]
a & = & 15 \tan(24^\circ) & \mbox{Find the exact value}\\[2ex]
a & \approx & 6.678 & \mbox{Use a calculator to approximate the answer}
\end{array}
\]
\end{example}
\subsection{Applications}
Trigonometry is a major tool for surveyors, who measure and record the position and size of
land, trees, buildings,
etc.
\begin{example}
Suppose we want to know the height of a building, to the nearest foot.
From a distance of \(185\) feet from the base
of the building, we measure the angle that the line of sight makes with the top of the
building (see picture). In surveying, this angle is called the \textit{angle of elevation}.
\[\img{U4_6F6.png}{}{20em}{}\]
The height \(h\) is the opposite angle, and we know the adjacent (\(185\) ft), so we use the
tangent function, and solve for \(h\):
\[\begin{array}{rcl}
\tan(36^\circ)& = & \displaystyle \frac{h}{185} \\[2ex]
185 \tan(36^\circ) & = & h \\[2ex]
h & \approx & 134 \mbox{ ft}
\end{array}
\]
\end{example}
Problems
\problem
Find the following trig values, approximated to three decimal places.
\[
\begin{array}{cccc}
\mbox{a. } \sin(238^\circ) & \mbox{b. } \displaystyle \cos\left( \frac{3\pi}{5}\right)
& \mbox{c. } \tan 4 & \mbox{d. } \sin (1.75)
\end{array}
\]
\begin{sol}
\begin{enumerate}
\item Put the calculator in degree mode:
\[\fbox{\(\sin\)}( \hspace{1ex} \fbox{\(2\)} \hspace{1ex}\fbox{\(3\)} \hspace{1ex} \fbox{\(8\)} \hspace{1ex}
\fbox{)}\hspace{1ex} \fbox{\(=\)} \hspace{1ex} -0.848 \]
\item
Put the calculator in radian mode:
\[\fbox{\(\cos\)}( \hspace{1ex} \fbox{\(3\)} \hspace{1ex}\fbox{\(\pi\)} \hspace{1ex} \fbox{\(\div\)} \hspace{1ex}
\fbox{\(5\)}\hspace{1ex} \fbox{)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} -0.309 \]
\item
In radian mode:
\[\fbox{\(\tan\)}( \hspace{1ex} \fbox{\(4\)} \hspace{1ex}\fbox{)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} 1.158 \]
\item
In radian mode:
\[\fbox{\(\sin\)}( \hspace{1ex} \fbox{\(1\)} \hspace{1ex}\fbox{\(.\color{white}{|}\)} \hspace{1ex} \fbox{\(7\)} \hspace{1ex} \fbox{\(5\)} \hspace{1ex} \fbox{)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} 0.984 \]
\end{enumerate}
\end{sol} \mproblem
Find the following trig values, approximated to three decimal places.
\[
\begin{array}{cccc}
\mbox{a. } \cos(45) & \mbox{b. } \displaystyle \sin\left( \frac{3}{5}\right) & \mbox{c. }
\tan (22.5^\circ) & \mbox{d. } \displaystyle \cos\left( \frac{11\pi}{15}\right)
\end{array}
\]
\problem
Find the following trig values, approximated to three decimal places.
\[
\begin{array}{cccc}
\mbox{a. } \sec (20^\circ) & \mbox{b. } \displaystyle \csc\left( \frac{5\pi}{12}\right)
& \mbox{c. } \cot (41^\circ) & \mbox{d. } \sec (2.25)
\end{array}
\]
\begin{sol}
\begin{enumerate}
\item Put the calculator in degree mode:
\[\fbox{\(\cos\)}( \hspace{1ex} \fbox{\(2\)} \hspace{1ex}\fbox{\(0\)} \hspace{1ex} \fbox{)} \hspace{1ex}
\fbox{\(=\)}\hspace{1ex} \fbox{\(x^{-1}\)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} 1.064 \]
\item
Put the calculator in radian mode:
\[\fbox{\(\sin\)}( \hspace{1ex} \fbox{\(5\)} \hspace{1ex}\fbox{\(\pi\)} \hspace{1ex} \fbox{\(\div\)} \hspace{1ex}
\fbox{1}\hspace{1ex} \fbox{2} \hspace{1ex} \fbox{)} \hspace{1ex} \fbox{\(=\)}
\hspace{1ex} \fbox{\(x^{-1}\)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} 1.035\]
\item
In degree mode:
\[\fbox{\(\tan\)}( \hspace{1ex} \fbox{\(4\)} \hspace{1ex}\fbox{\(1\)} \hspace{1ex} \fbox{)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} \fbox{\(x^{-1}\)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} 1.150 \]
\item
In radian mode:
\[\fbox{\(\cos\)}( \hspace{1ex} \fbox{\(2\)} \hspace{1ex}\fbox{\(.\color{white}{|}\)} \hspace{1ex} \fbox{\(2\)} \hspace{1ex} \fbox{\(5\)} \hspace{1ex} \fbox{)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} \fbox{\(x^{-1}\)} \hspace{1ex} \fbox{\(=\)} \hspace{1ex} -1.592 \]
\end{enumerate}
\end{sol}
\mproblem
Find the following trig values, approximated to three decimal places.
\[
\begin{array}{cccc}
\mbox{a. } \csc 1 & \mbox{b. } \displaystyle \cot\left( \frac{7\pi}{18}\right) &
\mbox{c. } \sec (87.5^\circ) & \mbox{d. } \cot (3.14)
\end{array}
\]
\problem
Find all sides in the triangle. Approximate the answer to three decimal places.
\[\img{U4_6F3.png}{}{20em}{}\]
\begin{sol}
Label the sides:
\[\img{U4_6F4.png}{}{20em}{}\]
Since we are given the hypotenuse, we can use the sine function to find the opposite, and
the cosine function to find the adjacent.
\[\begin{array}{rcll}
\sin(65^\circ)& = & \displaystyle \frac{b}{37} & \\[1ex]
37 \sin(65^\circ) & = & b & \\[1ex]
b &= &\displaystyle 37\sin(65^\circ) & \mbox{Exact answer}\\[1ex]
b & \approx & 33.533 & \mbox{Approximate answer}\\[1ex]
\cos(65^\circ)& = & \displaystyle \frac{a}{37} & \\[1ex]
37 \cos(65^\circ) & = & a & \\[1ex]
a &= &\displaystyle 37\cos(65^\circ) & \mbox{Exact answer}\\[1ex]
a & \approx & 15.637 & \mbox{Approximate answer}
\end{array}
\]
\end{sol}
\mproblem
Find all sides in the triangle. Approximate the answer to three decimal places.
\[\img{U4_6F5.png}{}{12em}{}\]
\problem
The angle of elevation to the top of a large Christmas tree is \(28^\circ\) when measured from a point
on the ground \(55\) ft from the base of the tree. Find the height of the tree.
\begin{sol}
Draw a picture:
\[\img{CTree.png}{}{25em}{}\]
Using the tangent function, we find:
\[\begin{array}{rcl}
\tan(28^\circ)& = & \displaystyle \frac{h}{55} \\[2ex]
55 \tan(28^\circ) & = & h \\[2ex]
h & \approx & 29 \mbox{ ft}
\end{array}
\]
\end{sol} \mproblem
The angle of elevation to the top of a water tower is measured to be \(71^\circ\) from a
point on the ground \(120\) ft from the base of the tower. Find the height of the water tower,
approximated to the nearest foot.