\begin{equation*}
\tan x = \frac{\sin x}{\cos x}
\end{equation*}
Issues arise when the denominator is zero, i.e., when \(\cos x=0\text{.}\) This leads to undefined points at \(x=\ldots, -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots\text{.}\) In general, any angle of the form \(n\frac{\pi}{2}\text{,}\) where \(n\) is an odd integer, should be excluded from the domain since tangent is undefined at these values.
The denominator becomes zero when \(\sin x=0\text{,}\) corresponding to \(x=\ldots,-\pi,0,\pi,2\pi,\ldots\text{.}\) In general, \(\cot x\) is undefined for angles of the form \(n\pi\text{,}\) where \(n\) is an integer. These angles should be excluded from the domain of cotangent.
Remark2.2.1.Domains for Tangent and Cotangent.
The domains for tangent and cotangent functions are given in Table 2.2.2
Table2.2.2.Domains of the tangent and cotangent functions
Function
Domain
\(\tan\theta\)
All real numbers except odd multiples of \(\frac{\pi}{2}\) (\(90^{\circ}\))
\(\cot\theta\)
All real numbers except integer multiples of \(\pi\) (\(180^{\circ}\))
Subsection2.2.2Ranges of the Tangent and Cotangent Functions
To determine the range of the tangent function, consider the point \(P(x, y)\) on the unit circle corresponding to the angle \(\theta\text{,}\) and let \(a\) be a real number such that \(a = \tan\theta = \frac{y}{x}\text{.}\)
Multiplying both sides by \(x\text{,}\) we obtain:
\begin{equation*}
y = ax
\end{equation*}
Squaring both sides yields:
\begin{equation*}
y^2 = a^2x^2
\end{equation*}
Substituting into the Pythagorean Identity (Definition 1.5.20), we have:
In other words, since \(a\) can be any real number and \(\tan\theta = a\text{,}\) the range of the tangent function consists of all real numbers. A similar method can be used to show that the range of the cotangent function is also the set of all real numbers.
Remark2.2.3.Ranges for Tangent and Cotangent.
The ranges for tangent and cotangent functions are given in Table 2.2.4
Table2.2.4.Ranges of the tangent and cotangent functions
Function
Range
\(\tan\theta\)
All real numbers
\(\cot\theta\)
All real numbers
Subsection2.2.3Domains of the Cosecant and Secant Functions
Consider the Reciprocal Identity for the cosecant function (Definition 1.4.2):
When \(\sin x=0\text{,}\) corresponding to \(x=\ldots,-\pi,0,\pi,2\pi,\ldots\text{,}\) the denominator becomes zero. In general, \(\csc x\) is undefined for angles of the form \(n\pi\text{,}\) where \(n\) is an integer, and these values should be excluded from the domain.
Similarly, since the cosecant function is defined as
we see that \(\sec x\) is undefined when \(\cos x=0\text{.}\) This occurs at \(x=\ldots, -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots\text{,}\) and thus any angle of the general form \(n\frac{\pi}{2}\text{,}\) where \(n\) is an odd integer, should be excluded from the domain of \(\sec x\text{.}\)
Remark2.2.5.Domains for Cosecant and Secant.
The domains for cosecant and secant functions are given in Table 2.2.6
Table2.2.6.Domains of the trigonometric functions
Function
Domain
\(\csc\theta\)
All real numbers except integer multiples of \(\pi\) (\(180^{\circ}\))
\(\sec\theta\)
All real numbers except integer multiples of \(\frac{\pi}{2}\) (\(90^{\circ}\))
Subsection2.2.4Ranges of the Cosecant and Secant Functions
If the angle is not an integer multiple of \(\pi\text{,}\) i.e. \(x\neq n\pi\text{,}\) where \(n\) is an integer, then the Reciprocal Identity allows us to define cosecant as
Therefore, taking the reciprocal of the range of sine, we get
\begin{equation*}
\csc x\leq-1\mbox{ or }\csc x\geq1\text{.}
\end{equation*}
In other words, the range of the cosecant function is all real numbers less than or equal to \(-1\) or greater than or equal to \(1\text{.}\)
Similarly, since \(-1\leq \cos x\leq1\text{,}\) we can get the range for the secant funcation as
\begin{equation*}
\sec x\leq-1\mbox{ or }\sec x\geq1
\end{equation*}
or all real numbers less than or equal to \(-1\) or greater than or equal to \(1\text{.}\)
Remark2.2.7.Ranges for Cosecant and Secant.
The ranges for cosecant and secant functions are given in Table 2.2.8
Table2.2.8.Ranges of the cosecant and secant functions
Function
Range
\(\csc\theta\)
All real numbers greater than or equal to \(1\) or less than or equal to \(-1\)
\(\sec\theta\)
All real numbers greater than or equal to \(1\) or less than or equal to \(-1\)
Subsection2.2.5The Tangent Function
In Subsection 1.5.4, we learned that the tangent function is periodic with a period of \(\pi\text{.}\) To graph \(y = \tan x\text{,}\) we focus on plotting one period and then repeat those values to complete the graph.
We also know that the domain of tangent includes all real numbers except any angle of the form \(n\frac{\pi}{2}\text{,}\) where \(n\) is an odd integer. These values are excluded since tangent is undefined there. In fact, any line of the form \(x=n\frac{\pi}{2}\) (e.g. \(x=-\frac{\pi}{2}\) and \(x=\frac{\pi}{2}\)) is a vertical asymptote
Knowing the location of the vertical asymptotes, we choose the interval \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) to plot our points for tangent. This interval has a length of \(\pi\) (one period), allowing us to repeat the values to complete the graph of tangent over the entire domain.
Recall from Definition 1.3.2 that on the unit circle, the expression \(\tan\theta=\frac{y}{x}\) denotes the ratio of the \(y\)-coordinate to the \(x\)-coordinate of a point \(P(x, y)\) associated with an angle \(\theta\text{.}\) This ratio changes dynamically as \(\theta\) varies from zero to \(\frac{\pi}{2}\text{.}\)
As \(\theta\) approaches zero, the \(y\)-value tends to zero, and \(x\) approaches 1, resulting in \(\tan\theta\) being a small fraction. Conversely, as \(\theta\) approaches \(\frac{\pi}{2}\text{,}\) the \(y\)-value approaches 1, while \(x\) becomes extremely small and near zero. This causes \(\tan\theta\) to evaluate as a fraction divided by a very small number, producing a large number. A similar effect occurs when \(\theta\) is between \(-\frac{\pi}{2}\) and zero, except \(\tan\theta\) is negative in this range. This behavior is shown in Figure 2.2.9.
Figure2.2.9.As \(\theta\) moves from \(-90^{\circ}\) to \(90^{\circ}\text{,}\) this figure plots the values of \(y = \tan \theta\text{.}\) Move the slider for \(\theta\) to see how changing the angle affects \(\tan\theta\text{.}\) Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.
Next we recall values of tangent for known angles, which are listed in Table 2.2.10 and plotted in Figure 2.2.11.
Figure2.2.11.The values in Table 2.2.10 plotted on a graph with a smooth curve connecting the points to make the curve for \(y=\tan(x)\text{.}\)
Notice the symmetrical nature of the tangent function’s graph with respect to the origin, a feature explained in Section 1.5.7, where we learned that tangent is an odd function.
Since the graph in Figure 2.2.11 represents one period, we can complete the graph of \(y=\tan x\) by extending the pattern in both directions to obtain Figure 2.2.12.
Figure2.2.12.The plot for \(y=\tan(x)\)
Subsection2.2.6The Cotangent Function
The domain of the cotangent function is defined for all angles except those of the form \(n\pi\text{,}\) where \(n\) is an integer. These excluded values correspond to vertical asymptotes. In fact, any line of the form \(x = n\pi\text{,}\) where \(n\) is an integer, serves as a vertical asymptote. Additionally, as we learned in Subsection 1.5.4, the cotangent function has a period of \(\pi\text{.}\) We can construct a plot for it in a manner similar to how we constructed the tangent function, as illustrated in Figure 2.2.13.
Figure2.2.13.As \(\theta\) moves from \(0^{\circ}\) to \(180^{\circ}\text{,}\) this figure plots the values of \(y = \cot \theta\text{.}\) Move the slider for \(\theta\) to see how changing the angle affects \(\cot\theta\text{.}\) Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.
Plotting points for \(y=\cot x\) and using the fact that the cotangent function is periodic, we obtain the graph for cotangent in Figure 2.2.14
Figure2.2.14.The plot for \(y=\cot(x)\text{.}\)
The symmetry of the cotangent function about the origin is evident, confirming its nature as an odd function, as explained in Section 1.5.7.
Subsection2.2.7The Cosecant Function
The cosecant function has vertical asymptotes at points \(n\pi\text{,}\) where \(n\) is an integer, corresponding to the values where the function is undefined. These points are the same ones excluded from the domain of \(\csc x\text{.}\) As discussed in Subsection 1.5.4, the cosecant function has a period of \(2\pi\text{.}\) Given this, we choose to examine its behavior within one period, specifically from \(0\) to \(2\pi\text{,}\) since this interval spans one complete period of the cosecant function.
Consider the Reciprocal Identity for the cosecant function (Definition 1.4.2):
As \(x\) approaches zero, sine decreases to zero, making cosecant approach positive infinity. Increasing \(x\) towards \(\frac{\pi}{2}\text{,}\)\(\sin(x)\) increases to \(1\text{,}\) and cosecant decreases to \(1\text{.}\) As \(x\) moves from \(\frac{\pi}{2}\) to \(\pi\text{,}\) sine approaches zero, causing \(\csc(x)\) to approach infinity.
Similarly, for \(x\gt\pi\) nearing \(\pi\text{,}\)\(\sin(x)\) becomes a small, negative number near zero, resulting in \(\csc(x)\) approaching negative infinity. As \(x\) increases to \(\frac{3\pi}{2}\text{,}\)\(\sin(x)\) decreases to \(-1\text{,}\) and \(\csc(x)\) increases to \(-1\text{.}\) Finally, from \(\frac{3\pi}{2}\) to \(2\pi\text{,}\)\(\sin(x)\) approaches a small negative number, causing cosecant to approach negative infinity. This behavior is shown in Figure 2.2.15.
Figure2.2.15.As \(\theta\) moves from \(0^{\circ}\) to \(360^{\circ}\text{,}\) this figure plots the values of \(y=\sin x\) and \(y = \csc \theta\text{.}\) Move the slider for \(\theta\) to see how changing the angle affects \(\csc\theta\text{.}\) Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.
Next we recall values of sine and cosecant for known angles, which are listed in Table 2.2.16 and plotted in Figure 2.2.17.
Table2.2.16.Values for \(y=\csc x\)
\(x\)
\(\sin x\)
\(\csc x\)
\(x\)
\(\sin x\)
\(\csc x\)
\(0\)
\(0\)
Undefined
\(\pi\)
\(0\)
Undefined
\(\frac{\pi}{6}\)
\(\frac{1}{2}\)
\(2\)
\(\frac{7\pi}{6}\)
\(-\frac{1}{2}\)
\(-2\)
\(\frac{\pi}{4}\)
\(\frac{\sqrt{2}}{2}\)
\(\sqrt{2}\)
\(\frac{5\pi}{4}\)
\(-\frac{\sqrt{2}}{2}\)
\(-\sqrt{2}\)
\(\frac{\pi}{3}\)
\(\frac{\sqrt{3}}{2}\)
\(\frac{2\sqrt{3}}{3}\)
\(\frac{4\pi}{3}\)
\(-\frac{\sqrt{3}}{2}\)
\(-\frac{2\sqrt{3}}{3}\)
\(\frac{\pi}{2}\)
\(1\)
\(1\)
\(\frac{3\pi}{2}\)
\(-1\)
\(-1\)
\(\frac{2\pi}{3}\)
\(\frac{\sqrt{3}}{2}\)
\(\frac{2\sqrt{3}}{3}\)
\(\frac{5\pi}{3}\)
\(-\frac{\sqrt{3}}{2}\)
\(-\frac{2\sqrt{3}}{3}\)
\(\frac{3\pi}{4}\)
\(\frac{\sqrt{2}}{2}\)
\(\sqrt{2}\)
\(\frac{7\pi}{4}\)
\(-\frac{\sqrt{2}}{2}\)
\(-\sqrt{2}\)
\(\frac{5\pi}{6}\)
\(\frac{1}{2}\)
\(2\)
\(\frac{11\pi}{6}\)
\(-\frac{1}{2}\)
\(-2\)
\(2\pi\)
\(0\)
Undefined
Figure2.2.17.The values in Table 2.2.16 plotted on a graph with a smooth curve connecting the points to make the curve for \(y=\csc(x)\text{.}\)
Since the graph in Figure 2.2.17 represents one period, we can complete the graph of \(y=\csc x\) by extending the pattern in both directions to obtain Figure 2.2.18.
Figure2.2.18.The plot for \(y=\sin(x)\) and \(y=\csc(x)\text{.}\)
Notice the graph of the cosecant function is symmetric with respect to the origin, confirming what we learned in Section 1.5.7, that cosecant is an odd function.
Subsection2.2.8The Secant Function
As discussed earlier in this section, the secant function has a domain for all real numbers except angles of \(n\frac{\pi}{2}\text{,}\) where \(n\) is an integer. These excluded values correspond to the vertical asymptotes of the secant function. With a period of \(2\pi\text{,}\) we can focus on the interval \(0\) to \(2\pi\text{.}\) The construction of the secant function plot follows a similar approach to that used for the cosecant function, as illustrated in Figure 2.2.19.
Figure2.2.19.As \(\theta\) moves from \(0^{\circ}\) to \(360^{\circ}\text{,}\) this figure plots the values of \(y=\cos x\) and \(y = \sec \theta\text{.}\) Move the slider for \(\theta\) to see how changing the angle affects \(\sec\theta\text{.}\) Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.
Plotting points for \(y=\sec x\) and using the fact that the secant function is periodic, we obtain the graph for secant in Figure 2.2.20.
Figure2.2.20.The plot for \(y=\cos(x)\) and \(y=\sec(x)\text{.}\)
Notice the graph of the secant function is symmetric about the \(y\)-axis, and thus secant is an even function, confirming what we learned in Section 1.5.7.
Subsection2.2.9Graphing Transformations of Other Trigonometric Functions
Similar to the graphs of sine and cosine, the graphs of the other trigonometric functions can undergo vertical stretching and compressing, horizontal stretching and compressing, phase shifts, vertical shift transformations, and reflections about the \(x\)- and \(y\)-axes. However, unlike the sine and cosine functions, there is no amplitude for the other trigonometric functions. These transformations are listed in Definition 2.2.21.
Definition2.2.21.Transformations of the Tangent, Cotangent, Cosecant, and Secant Functions.
For functions of the form
\begin{equation*}
y = A \cdot \tan(B(x - C)) + D, \quad y = A \cdot \cot(B(x - C)) + D,
\end{equation*}
\begin{equation*}
\quad y = A \cdot \csc(B(x - C)) + D, \quad \mbox{and} \quad y = A \cdot \sec(B(x - C)) + D\text{,}
\end{equation*}
we can express the transformations as follows:
Vertical Compression/Stretch: \(|A|\)
\(|A|\) is the value of the vertical stretch/compression.
If \(|A| \gt 1\text{,}\) there is vertical stretching.
If \(0 \lt |A| \lt 1\text{,}\) there is vertical compression.
Period and Horizontal Stretch/Compression: \(|B|\)
The period is \(\frac{\pi}{|B|}\) for tangent and cotangent, and \(\frac{2\pi}{|B|}\) for cosecant and secant.
If \(|B| \gt 1\text{,}\) there is horizontal compression, and the period is shortened.
If \(0 \lt |B| \lt 1\text{,}\) there is horizontal stretching, and the period is lengthened.
Phase Shift: \(C\)
If \(C\) is positive, there is a shift to the right.
If \(C\) is negative, there is a shift to the left.
Vertical Shift: \(D\)
If \(D\) is positive, there is a shift upward.
If \(D\) is negative, there is a shift downward.
Reflection about the \(x\)-axis:
If \(A\) is negative (\(A \lt 0\)), there is a reflection about the \(x\)-axis.
Reflection about the \(y\)-axis:
If \(B\) is negative (\(B \lt 0\)), there is a reflection about the \(y\)-axis.
Vertical Asymptotes:
For tangent and cotangent, vertical asymptotes occur at
\begin{equation*}
x = C + \frac{\pi}{|B|}n\text{,}
\end{equation*}
where \(n\) is an integer.
For cosecant and secant, vertical asymptotes occur at
\begin{equation*}
x = C + \frac{\pi}{2|B|}n\text{,}
\end{equation*}
where \(n\) is an integer.
Remark2.2.22.Other Forms of Transformations.
For functions of the form
\begin{equation*}
y = A \cdot \tan(Bx - E) + D, \quad y = A \cdot \cot(Bx - E) + D,
\end{equation*}
\begin{equation*}
\quad y = A \cdot \csc(Bx - E) + D, \quad \mbox{and} \quad y = A \cdot \sec(Bx - E) + D\text{,}
\end{equation*}
the transformations are the same as above, except for the phase shift and vertical asymptotes where you replace \(C\) with \(\frac{E}{B}\text{.}\) If \(\frac{E}{B} \lt0\) the phase shift is to the right, and if \(\frac{E}{B} \gt 0\) it is to the left.
For tangent and cotangent, vertical asymptotes occur at
\begin{equation*}
x = \frac{E}{B} + \frac{\pi}{|B|}n\text{,}
\end{equation*}
where \(n\) is an integer.
For cosecant and secant, vertical asymptotes occur at
\begin{equation*}
x = \frac{E}{B} + \frac{\pi}{2|B|}n\text{,}
\end{equation*}
where \(n\) is an integer.
Example2.2.23.Vertical Stretch/Compression and Reflection about the \(x\)-axis.
Graph each function
\(\displaystyle y=\tan(x)\)
\(\displaystyle y=2\tan(x)\)
\(\displaystyle y=\frac{1}{2}\tan(x)\)
\(\displaystyle y=-\tan(x)\)
Solution.
Figure2.2.24.The transformations of the tangent function graph, starting with a baseline of \(y=\tan(x)\) along with graphs with a vertical stretch, a vertical compression, and a reflection about the \(x\)-axis.
Example2.2.25.Horizontal Stretch/Compression and Reflection about the \(y\)-axis.
Identify the period and graph one period for each of the following functions:
Figure2.2.26.The transformations of the cotangent function graph, starting with a baseline of \(y=\cot(x)\) along with graphs with a horizontal stretch, a horizontal compression, and a reflection about the \(y\)-axis.
Example2.2.27.Phase Shifts and Vertical Shifts.
Graph the function \(y=\csc\left(x-\frac{\pi}{2}\right)+2\text{.}\)
Solution.
Figure2.2.28.The graphs of \(y=\csc\left(x-\frac{\pi}{2}\right)+2\) and \(y=\sin\left(x-\frac{\pi}{2}\right)+2\) start with the plots of \(y=\csc(x)\) and \(y=\sin(x)\) and add a phase shift of \(\frac{\pi}{2}\) to the right and a vertical shift up by \(2\text{.}\)
Explore the effects of various transformations using the interactive features in Figure 2.2.29 and Figure 2.2.30.
Figure2.2.29.Manipulate the graphs of tangent and cotangent by adjusting the sliders for \(A\text{,}\)\(B\text{,}\)\(C\text{,}\) and \(D\text{.}\) Observe the effects on period, phase and vertical shifts, as well as reflections about the \(x\)- and \(y\)-axes. Additionally, you can toggle between the tangent and cotangent graphs by selecting the corresponding function.Figure2.2.30.Manipulate the graphs of cosecant and secant by adjusting the sliders for \(A\text{,}\)\(B\text{,}\)\(C\text{,}\) and \(D\text{.}\) Observe the effects on period, phase and vertical shifts, as well as reflections about the \(x\)- and \(y\)-axes. Additionally, you can toggle between the cosecant and secant graphs by selecting the corresponding function.
When navigating the open ocean, maintaining a straight course poses challenges due to limited visual markers. One technique involves the steersperson using the positions of shadows cast by objects on the canoe—such as crew members, railings, and sails—to keep them fixed on the deck, ensuring a straight trajectory. However, if the canoe veers off course, the changing position of the canoe relative to the sun leads to a shift in the shadows. Observing these shadow movements allows the steersperson to make course corrections. It’s important to note that this method is effective only over a short duration, as the sun’s continuous movement across the sky causes ongoing changes in shadow positions. To illustrate the limitations over extended periods, consider the example of the Samoan double-hulled voyaging vaʻa, Gaualofa, with a 14-meter-high mast. The length of the shadow is modeled by
where \(l\) is the shadow length in meters and \(t\) represents the hours since 6 am (assuming sunrise at 6 am and sunset at 6 pm). In each of the following questions, calculate the length of the shadow, rounded to the nearest tenth of a meter, for the given time.
An observer on Rangiroa spots the Faʻafaite, a double-hulled voyaging canoe from Tahiti, sailing off the north coast of the atoll, maintaining a distance of three nautical miles from the shore and traveling east. Let \(\theta\) represent the angle formed between the line from the observer to the vaʻa and a line extending due north from the observer, measured in radians. The angle \(\theta\) is negative if the vaʻa is to the left of the observer and positive when to the right, as shown in the figure above. The distance (in nautical miles), denoted by \(d(\theta)\) from Faʻafaite to the observer is given by the function
In each of the following questions, calculate the distance from the observer to Faʻafaite, \(d(\theta)\text{,}\) in nautical miles, for the given angle \(\theta\text{.}\) Round your answer to two decimal places.
