\(\tan x\) is undefined when the denominator is zero, that is, 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.
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 \(a = \tan\theta = \frac{y}{x}\text{.}\)
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.
When \(\sin x=0\text{,}\) corresponding to \(x=\ldots,-\pi,0,\pi,2\pi,\ldots\text{,}\) the denominator becomes zero. Therefore, \(\csc x\) is undefined for angles of the form \(n\pi\text{,}\) where \(n\) is an integer, and these values should be excluded from the domain.
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 form \(n\frac{\pi}{2}\text{,}\) where \(n\) is an odd integer, should be excluded from the domain of \(\sec x\text{.}\)
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 cosecant is defined by the Reciprocal Identity as
In SubsectionΒ 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 the graph for one period and then repeating those values to complete the graph.
We also know that the domain of tangent includes all real numbers except angles of the form \(n\frac{\pi}{2}\text{,}\) where \(n\) is an odd integer. These values are excluded since tangent is undefined for these angles. In fact, any line of the form \(x=n\frac{\pi}{2}\text{,}\) where \(n\) is an odd integer (e.g., \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\)), 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 its entire domain.
Recall from DefinitionΒ 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 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 approaches zero. This causes \(\tan\theta\) to be evaluated as a fraction with a very small number in the denominator, 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 FigureΒ 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. If you are viewing the PDF or a printed copy, you can scan the QR code or follow the βStandaloneβ link to explore the interactive version online.
Notice the symmetrical nature of the tangent functionβs graph with respect to the origin, a feature explained in SectionΒ 1.5.7, where we learned that tangent is an odd function.
Since the graph in FigureΒ 2.2.11 represents one period, we can complete the graph of \(y=\tan x\) by extending the pattern in both directions to obtain FigureΒ 2.2.12.
The domain of the cotangent function contains 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 SubsectionΒ 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 FigureΒ 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. If you are viewing the PDF or a printed copy, you can scan the QR code or follow the βStandaloneβ link to explore the interactive version online.
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 SubsectionΒ 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.
As \(x\) approaches zero, \(\sin x\) decreases to zero, making \(\csc x\) approach positive infinity. Increasing \(x\) towards \(\frac{\pi}{2}\text{,}\)\(\sin(x)\) increases to \(1\text{,}\) and cosecant decreases to \(1\text{.}\) As \(x\) increases from \(\frac{\pi}{2}\) to \(\pi\text{,}\)\(\sin x\) approaches zero, causing \(\csc(x)\) to approach infinity.
Similarly, for \(x\gt\pi\) but 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, as \(x\) increases from \(\frac{3\pi}{2}\) to \(2\pi\text{,}\)\(\sin(x)\) approaches zero from the negative side, causing cosecant to approach negative infinity. This behavior is shown in FigureΒ 2.2.15.
Figure2.2.15.As \(\theta\) moves from \(0^{\circ}\) to \(360^{\circ}\text{,}\) this figure plots the values of \(y=\sin \theta\) 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. If you are viewing the PDF or a printed copy, you can scan the QR code or follow the βStandaloneβ link to explore the interactive version online.
Since the graph in FigureΒ 2.2.17 represents one period, we can complete the graph of \(y=\csc x\) by extending the pattern in both directions to obtain FigureΒ 2.2.18.
Notice the graph of the cosecant function is symmetric with respect to the origin, confirming what we learned in SectionΒ 1.5.7, that cosecant is an odd function.
As discussed earlier in this section, the secant function has a domain of all real numbers except angles of the form \(n\frac{\pi}{2}\text{,}\) where \(n\) is an odd 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 graph follows a similar approach to the one used for the cosecant function, as illustrated in FigureΒ 2.2.19.
Figure2.2.19.As \(\theta\) moves from \(0^{\circ}\) to \(360^{\circ}\text{,}\) this figure plots the values of \(y=\cos \theta\) 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. If you are viewing the PDF or a printed copy, you can scan the QR code or follow the βStandaloneβ link to explore the interactive version online.
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 SectionΒ 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 DefinitionΒ 2.2.21.
\begin{equation*}
y = A \tan(Bx - E) + D, \quad y = A \cot(Bx - E) + D\text{,}
\end{equation*}
\begin{equation*}
\quad y = A \csc(Bx - E) + D, \quad \mbox{and} \quad y = A \sec(Bx - E) + D\text{,}
\end{equation*}
the transformations are the same as above, except for the phase shift and vertical asymptotes where we replace \(C\) with \(\frac{E}{B}\text{.}\) If \(\frac{E}{B} \gt0\) the phase shift is to the right, and if \(\frac{E}{B} \lt 0\) it is to the left.
Figure2.2.24.The transformations of the tangent function graph, starting with the graph of \(y=\tan(x)\) along with graphs with a vertical stretch, a vertical compression, and a reflection about the \(x\)-axis.
Figure2.2.26.The transformations of the cotangent function graph, starting with the graph of \(y=\cot(x)\) along with graphs with a horizontal stretch, a horizontal compression, and a reflection about the \(y\)-axis.
Figure2.2.28.The graphs of \(y=\csc\left(x-\frac{\pi}{2}\right)+2\) and \(y=\sin\left(x-\frac{\pi}{2}\right)+2\) are derived from the graphs of \(y=\csc(x)\) and \(y=\sin(x)\) by applying a phase shift of \(\frac{\pi}{2}\) to the right and a vertical shift up by \(2\text{.}\)
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, toggle between the tangent and cotangent graphs by selecting the corresponding function. If you are viewing the PDF or a printed copy, you can scan the QR code or follow the βStandaloneβ link to explore the interactive version online.
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, toggle between the cosecant and secant graphs by selecting the corresponding function. If you are viewing the PDF or a printed copy, you can scan the QR code or follow the βStandaloneβ link to explore the interactive version online.
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 canoeβsuch as crew members, railings, and sailsβto 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. Itβs important to note that this method is effective only over a short duration, as the sunβs 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 vaΚ»a, 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 FaΚ»afaite, a double-hulled voyaging canoe from Tahiti, sailing off the north coast of the atoll, maintaining a distance of 1.5 kilometers from the shore and traveling east. Let \(\theta\) represent the angle formed between the line from the observer to the vaΚ»a and a line extending due north from the observer, measured in radians. The angle \(\theta\) is negative if the vaΚ»a is to the left of the observer and positive when to the right, as shown in the figure above. The distance (in kilometers), denoted by \(d(\theta)\) from FaΚ»afaite to the observer is given by the function
In each of the following questions, calculate the distance from the observer to FaΚ»afaite, \(d(\theta)\text{,}\) in kilometers, for the given angle \(\theta\text{.}\) Round your answer to two decimal places.
