Sunrise and sunset are vital markers for navigational orientation and course corrections. During sunrise, you can determine the direction and observe the origin of wind and waves. As the sun rises higher, steering by the sun becomes impractical, and you must instead depend on swells to keep your course. At sunset, you can reassess your position and take note of any changes in wind and swell patterns. Although the general pattern is for the sun to rise in the east and set in the west, its exact position on the horizon, known as solar declination, varies throughout the year.
To understand why the position changes, we must first learn about Earth’s axial tilt, which represents the angle between the Earth’s rotational axis and its orbital plane around the Sun. To conceptualize this tilt, envision a pole passing through the Earth’s center, extending from the North to South Pole, with the Earth revolving around this axis. Each complete rotation of the Earth on this axis corresponds to one day. As the Earth travels along its orbit around the Sun with a constant tilt, the orientation of the Earth’s axial tilt causes either the North Pole or the South Pole to tilt toward the Sun. This tilt varies depending on the Earth’s location in its orbit relative to the Sun.
During the equinoxes, which mark the transition from winter to spring and from summer to fall, the Earth’s axis is not tilted towards or away from the Sun. Consequently, on these days, the Sun rises due east and sets due west, and the duration of day and night is approximately equal.
Following the fall equinox in the Southern Hemisphere, typically occurring around March 20th, the Earth proceeds along its orbit around the Sun, leading the Southern Hemisphere to tilt away from the Sun. As a result, the Sun’s rising position progressively moves northward each day. By the time of the winter solstice, around June 20th, the Sun rises from its northernmost position, resulting in the shortest day in the Southern Hemisphere and the longest day of the year in the Northern Hemisphere.
After the winter solstice, the Earth’s axial tilt remains the same, but the Southern Hemisphere starts tilting toward the Sun. This causes the Sun’s rising position to gradually shift southward each day. When the spring equinox arrives, approximately on September 22nd, the Sun rises due east once again, and day and night are once more of equal duration.
As the Southern Hemisphere continues to tilt towards the Sun, the Sun’s rising position moves even further south each day. By the time of the summer solstice, approximately on December 21st, the Sun rises from its southernmost position, resulting in the longest day of the year in the Southern Hemisphere and the shortest day in the Northern Hemisphere.
Following the summer solstice, the Southern Hemisphere begins tilting away from the Sun, causing the Sun’s rising position to gradually shift northward each day. This movement continues until the next fall equinox, completing the annual cycle.
Figure2.1.1.Illustration shows the relative positions and timing of solstice, equinox and seasons in relation to the Earth’s orbit around the Sun and the axial tilt. During the June solstice, the northern hemisphere is tilted towards the Sun, while the southern hemisphere is tilted away. Conversely, during the December solstice, the southern hemisphere is tilted towards the Sun, and the northern hemisphere is tilted away. During the March and September equinoxes, the Earth’s axis is perpendicular to the line connecting the Earth to the Sun, causing the Sun to appear directly above the equator. As a result, day and night are approximately equal in length all over the world.
The position where the Sun rises throughout the year exhibits a repeating pattern, characterized as a periodic function with a cycle of about one year. This function can be mathematically represented as either a sine or cosine function. Graphically, it illustrates the sunrise position on the horizon relative to the east (Hikina) as time progresses. By examining this function, we can observe the gradual northward and southward movement of the sunrise position throughout the year. Key dates such as the equinoxes and solstices mark significant points in this pattern, providing insights into the changing seasons and variations in daylight hours.
In this section, we will explore the periodic nature of sine and cosine functions and study their transformations. Studying the graphs of sine and cosine functions provides valuable insights into the world around us. The graphs of the remaining trigonometric functions will be covered in Section 2.2.
Subsection2.1.1Domain and Range of Sine and Cosine
The domains for \(\sin(\theta)\) and \(\cos(\theta)\) consist of the set of the inputs of the functions. Since any angle \(\theta\) can be input into sine and cosine and still have these functions defined, the domain for both sine and coseine is all real numbers. Recall that in Section 1.3, if \(P(x,y)\) is any point on the unit circle that corresponds to the angle \(\theta\text{,}\) we defined \(\sin\theta=y\) and \(\cos\theta=x\text{.}\) Given the constraints on the unit circle, \(-1\leq x\leq1\) and \(-1\leq y\leq1\text{,}\) and thus
Since the range of a function consists of all its outputs, we conclude that the range of both the sine and cosine functions spans all real numbers between -1 and 1.
Remark2.1.2.Domain and Range for the Sine and Cosine Functions.
Table 2.1.3 summarizes the domains and ranges for the sine and cosine functions.
Table2.1.3.Domains and Ranges of Sine and Cosine
Function
Domain
Range
\(\sin\theta\)
All real numbers, \((-\infty,\infty)\)
All real numbers from -1 to 1, \([-1,1]\)
\(\cos\theta\)
All real numbers, \((-\infty,\infty)\)
All real numbers from -1 to 1, \([-1,1]\)
Subsection2.1.2The Sine Function
Convention2.1.4.
In Chapter 1, trigonometric functions typically use \(\theta\) or \(t\) as the variable in the domain, such as \(y=\cos\theta\) and \(y=\sin t\text{.}\) However, when graphing functions on the Cartesian plane (\(xy\)-coordinate system), \(x\) is conventionally used as the variable in the domain. Therefore, when graphing trigonometric functions, we will use \(x\) as the variable, for example \(y=\cos x\) and \(y=\sin x\text{.}\)
The sine function, as discussed in Subsection 1.5.4, is a periodic functions with period \(2\pi\text{.}\) To graph \(y=\sin x\text{,}\) we can focus on the interval \([0,2\pi]\text{.}\) By plotting this interval, we can then repeat the values over the entire domain to complete the remaining graph.
Recall from Definition 1.3.2 that on the unit circle, \(\sin \theta\) is defined to be the \(y\)-value of the terminal point \(P(x,y)\) on the unit circle associated with the angle \(\theta\text{.}\) As the angle increases from \(0\) to \(\frac{\pi}{2}\text{,}\) the \(y\)-value also increases from \(0\) to \(1\text{.}\) When the angle continues from \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\text{,}\) the \(y\)-value decreases from \(1\) to \(-1\text{.}\) Finally, as the angle approaches \(2\pi\text{,}\) the \(y\)-value increases to \(0\text{.}\) This behavior is shown in Figure 2.1.5.
Figure2.1.5.As \(\theta\) moves from \(0^{\circ}\) to \(360^{\circ}\text{,}\) this figure plots the values of \(y = \sin \theta\text{.}\) Move the slider for \(\theta\) to see how changing the angle affects \(\sin\theta\text{.}\) Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle. For example, you may not be aware that \(2\) radians (\(\approx114.6^{\circ}\)) lies in Quadrant II.
Next we recall known values for the sine function listed on Table 2.1.6.
Now that we have a visual understanding of the graph for \(y=\sin x\text{,}\) we can utilize the data from Table 2.1.6 to map out the points. This process enables us to construct the graph illustrated in Figure 2.1.7, representing one complete period of the sine function.
Figure2.1.7.The values in Table 2.1.6 plotted on a graph with a smooth curve connecting the points to make the curve for \(y=\sin(x)\text{.}\)
Since the graph in Figure 2.1.7 represents one period, we can now complete the graph of \(y=\sin x\) by extending the pattern in both directions to obtain Figure 2.1.8.
Figure2.1.8.The plot for \(y=\sin(x)\text{.}\)
Notice the graph of the sine function’s symmetry with respect to the origin, a characteristic supported in Section 1.5.7, where we learned that sine is an odd function.
Subsection2.1.3The Cosine Function
Similarly, we can construct a plot for the cosine function, shown in Figure 2.1.9.
Figure2.1.9.As \(\theta\) moves from \(0^{\circ}\) to \(360^{\circ}\text{,}\) this figure plots the values of \(y = \cos \theta\text{.}\) Move the slider for \(\theta\) to see how changing the angle affects \(\cos\theta\text{.}\) Note that while we will generally be using radians when graphing trigonometric functions, this figure uses degrees to help visualize the angle.
By plotting points for \(y=\cos x\) and using the fact that the cosine function is periodic, we obtain the graph for cosine over the entire domain. This is shown in Figure 2.1.10.
Figure2.1.10.The plot for \(y=\cos(x)\text{.}\)
In alignment with Section 1.5.7, observe that the graph of cosine is symmetric about the \(y\)-axis, confirming that it is an even function.
Definition2.1.11.
The graphs for the sine and cosine functions are commonly referred to as sinusoidal graphs or sinus curves.
Subsection2.1.4Graphing Transformations of Sine and Cosine
Now that we’ve become familiar with the graphs of the sine and cosine functions, let’s apply algebraic graphing techniques to these functions. Recall that when \(D \gt 0\text{,}\) the graph of \(y = f(x) + D\) shifts the graph of \(y = f(x)\) upward by \(D\) units, and the graph of \(y = f(x) - D\) shifts the graph of \(y = f(x)\) downward by \(D\) units.
depict the graphs of \(y = \sin(x)\) and \(y = \cos(x)\) with a downward vertical shift by \(D\) units, respectively.
Example2.1.13.Vertical Shifts.
Graph each function
\(\displaystyle y=\sin(x)+2\)
\(\displaystyle y=\cos(x)-1\)
Solution.
Figure2.1.14.The graph of \(y=\sin(x)+2\) is the same as the graph of \(y=\sin(x)\) but shifted up by 2 units.
Figure2.1.15.The graph of \(y=\cos(x)-1\) is the same as the graph of \(y=\cos(x)\) but shifted down by 1 unit.
Additionally, remember that the graph \(y = -f(x)\) reflects the graph of \(y = f(x)\) about the \(x\)-axis.
Definition2.1.16.Reflection about the \(x\)-axis.
The functions
\begin{equation*}
y=-\sin x \quad \mbox{and} \quad y=-\cos x
\end{equation*}
represent the graphs of \(y = \sin(x)\) and \(y = \cos(x)\) with a reflection about the \(x\)-axis, respectively.
Example2.1.17.Reflections about the \(x\)-axis.
Graph \(y=-\cos(x)\)
Solution.
Figure2.1.18.The graph \(y = -\cos(x)\) is obtained by multiplying every \(y-\)value of the \(y = \cos(x)\) graph by \(-1\text{.}\) This transformation reflects all points across the x-axis, turning positive values negative and negative values positive.
Similarly, recall that the graph of \(y=f(-x)\) reflects the graph of \(y=f(x)\) about the \(y\)-axis.
represent the graphs of \(y = \sin(x)\) and \(y = \cos(x)\) with a reflection about the \(y\)-axis, respectively.
Example2.1.20.Reflections about the \(y\)-axis.
Graph \(y=\sin(-x)+1\)
Solution.
Figure2.1.21.The graph \(y = \sin(-x) + 1\) is obtained by reflecting the graph of \(y = \sin(x)\) about the \(y\)-axis and then vertically shifting it upward by \(1\text{.}\) This transformation turns positive values of \(x\) negative and negative values positive, and increases every \(y\)-value by \(1\text{.}\)
Example2.1.22.Vertical Stretches and Compressions.
Graph each function
\(\displaystyle y=2\cos(x)\)
\(\displaystyle y=\frac{1}{2}\cos(x)\)
Solution.
Figure2.1.23.The graph of \(y = 2\cos(x)\) is achieved by vertically stretching the \(y\)-values of \(y = \cos(x)\) by a factor of 2. Similarly, the graph of \(y = \frac{1}{2}\cos(x)\) is obtained by vertically compressing the \(y\)-values of \(y = \cos(x)\) by a factor of \(\frac{1}{2}\text{.}\)
The factor multiplied at the front of the cosine function plays a crucial role in stretching and compressing the graph. This factor is known as the amplitude, measuring the maximum vertical distance from the midline to the peak or trough of a sinusoidal wave. In Example 2.1.22, the amplitude of \(y=2\cos(x)\) is 2, indicating a vertical stretch by a factor of 2 compared to the standard cosine function. Conversely, for \(y=\frac{1}{2}\cos(x)\text{,}\) the amplitude is \(\frac{1}{2}\text{,}\) which represented a vertical compression by a factor of \(\frac{1}{2}\text{.}\)
Definition2.1.24.Amplitude.
For a sinusoidal function, the amplitude, , denoted as \(|A|\text{,}\) is the height of the function, representing half the distance between its maximum and minimum values:
In other words, the amplitude is the vertical distance from the midline to the maximum or minimum value of the function. The midline is a horizontal line representing the average value of the function. It can be calculated by:
\begin{equation*}
y = \frac{\text{maximum} + \text{minimum}}{2}\text{.}
\end{equation*}
For a graph centered about the \(x\)-axis, the amplitude is simply the maximum value of the function.
In general, for
\begin{equation*}
y = A \cdot \sin(x) \quad \mbox{or} \quad y = A \cdot \cos(x)\text{,}
\end{equation*}
the amplitude is given by \(|A|\text{.}\) This absolute value ensures that amplitude is always a positive value, representing the magnitude of the vertical stretching or compression.
Definition2.1.25.Vertical Strech/Compression.
The functions
\begin{equation*}
y=A\sin x\quad\mbox{and}\quad y=A\cos x
\end{equation*}
represents a sine and consine function, respectively with an amplitude of \(|A|\text{.}\) The amplitude determines the vertical stretch or compression of the graph.
If \(|A| \gt 1\text{,}\) the graph undergoes a vertical stretch, making the peaks and troughs higher.
If \(0 \lt |A| \lt 1\text{,}\) the graph undergoes a vertical compression, reducing the distance between the peaks and troughs.
Example2.1.26.
Graph \(y=-4\sin x\) and identify the amplitude.
Solution.
The amplitude is \(|-4|=4\text{.}\)
Figure2.1.27.Since the amplitude of \(y = -4\sin(x)\) is \(4\text{,}\) the graph is stretched by a factor of 4 and will oscillate between \(-4\) and \(4\text{.}\) Additionally, the negative sign indicates that the graph is reflected about the \(x\)-axis.
You may recall from algebra that for functions of the form \(y=f(Bx)\text{,}\) a key factor emerges: when \(|B| \gt 1\text{,}\) the graph undergoes horizontal compression by a factor of \(\frac{1}{|B|}\text{;}\) conversely, when \(0 \lt |B| \lt 1\text{,}\) the graph is horizontally stretched by a factor of \(\frac{1}{|B|}\text{.}\) Given that sine and cosine complete one period in \(2\pi\text{,}\) the horizontal stretching or compressing of a period will be by a factor of \(\frac{1}{|B|}\text{.}\)
Figure2.1.31.One period each of \(y=\sin(2x)\text{,}\)\(y=\sin\left(\frac{1}{2}x\right)\text{,}\) and \(y=\sin\left(\frac{1}{3}x\right)\) compared to the standard \(y=\sin(x)\) graph. Observe the distinct effects of horizontal compression (when \(B=2\text{,}\) reducing the period to \(\frac{\pi}{2}\)) and stretching (when \(B=\frac{1}{2}\) and \(B=\frac{1}{3}\text{,}\) increasing the period to \(4\pi\) and \(6\pi\text{,}\) respectively).
Our final transformation involves functions of the form \(y = f(x - c)\text{.}\) When \(c \gt 0\text{,}\) the graph of \(y = f(x)\) is shifted \(c\) units to the right; when \(c \lt 0\text{,}\) it is shifted \(|c|\) units to the left.
Definition2.1.32.Phase Shift.
The functions
\begin{equation}
y = \sin(B(x - C)) \quad \mbox{and} \quad y = \cos(B(x - C))\tag{2.1.1}
\end{equation}
undergo a horizontal shift, known as phase shift, of \(C\) units. If \(C \gt 0\text{,}\) the phase shift is to the right; if \(C \lt 0\text{,}\) it is to the left.
Remark2.1.33.Functions of the form \(y = \sin(Bx - E)\) and \(y = \cos(Bx - E)\).
Note that you may see functions written in the form
\begin{equation}
y = \sin(Bx - E) \quad \mbox{and} \quad y = \cos(Bx - E).\tag{2.1.2}
\end{equation}
There is a subtle yet important difference between (2.1.1) and (2.1.2). In (2.1.1), the term \(B\text{,}\) affecting the period, is multiplied by both \(x\) and \(C\text{,}\) the phase shift. In (2.1.2), \(B\) is only multiplied by \(x\text{.}\) We can rewrite (2.1.2) by factoring out \(B\) as
\begin{equation*}
y = \sin\left(B\left(x - \frac{E}{B}\right)\right) \quad \mbox{and} \quad y = \cos\left(B\left(x - \frac{E}{B}\right)\right).
\end{equation*}
This form aligns with (2.1.1). Therefore, for equations of the form
\begin{equation*}
y = \sin(Bx - E) \quad \mbox{and} \quad y = \cos(Bx - E)
\end{equation*}
the phase shift is \(\frac{E}{B}\) units.
Example2.1.34.Phase Shift.
Identify the period and phase shift of each function, and graph the function
\begin{equation*}
\mbox{phase shift}=\frac{E}{B}=\frac{\pi}{1}=\pi \mbox{ (positive value indicates a shift to the right)}.
\end{equation*}
A positive value for the phase shift indicates a shift to the right. It’s important to note that, given \(B=1\text{,}\) the phase shift is simply \(E=\pi\text{.}\)
Figure2.1.35.The graph of \(y=\cos(x-\pi)\) is the graph of \(y=\cos x\) with a phase shift of \(\pi\) units to the right.
Since the given equation is in the form \(y = \sin\left(\frac{\pi}{6}(x + 2)\right)\text{,}\) we can identify \(B = \frac{\pi}{6}\) and \(C = -2\text{.}\) Consequently,
The negative sign indicates a phase shift to the left by 2 units.
To graph \(y=\sin\left(\frac{\pi}{6}(x+2)\right)\text{,}\) begin by graphing the sine function \(y=\sin\left(\frac{\pi}{6}x\right)\) with a period of 12. Then, apply a phase shift of 2 units to the left on the resulting graph.
Figure2.1.36.The graph of \(y=\sin\left(\frac{\pi}{6}x\right)\) represents the sine function \(y=\sin x\) with a horizontal stretch, resulting in a period of 12.
Figure2.1.37.Shifting the graph of \(y=\sin\left(\frac{\pi}{6}x\right)\) 2 units to the left results in the graph of \(y=\sin\left(\frac{\pi}{6}(x+2)\right)\text{.}\)
We will now summarize the transformations by consolidating them into a single equation.
the transformations are the same as above, except for the phase shift 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.
Explore the effects of various transformations using the interactive features in Figure 2.1.40.
Figure2.1.40.Manipulate the graphs of sine and cosine by adjusting the sliders for \(A\text{,}\)\(B\text{,}\)\(C\text{,}\) and \(D\text{.}\) Observe the effects on amplitude, period, phase and vertical shifts, as well as reflections about the \(x\)- and \(y\)-axes. Additionally, you can toggle between the sine and cosine graphs by selecting the corresponding function.
At the start of this section, we explored the effects of axial tilt on Earth’s seasons, considering the sun’s declination—the angle between the equator and a line drawn from the center of the Earth to the center of the Sun. When observing the sunrise, the sun’s declination is the angle from the sunrise to due east.
During the June Solstice, the declination is \(\delta=23.45^{\circ}\text{,}\) causing the sun to rise \(23.45^{\circ}\) to the north of east. On the December Solstice, the declination is \(\delta=-23.45^{\circ}\text{,}\) resulting in the sun rising \(23.45^{\circ}\) to the south of east.
Conversely, during the spring and fall equinox when the declination is \(\delta=0^{\circ}\text{,}\) the sun rises precisely at due east. This period holds significant historical importance, as observers, both in the past and present, have utilized this time to precisely determine the eastward direction from their positions. This practice is widespread across various cultures, with individuals using the equinox to establish directional markers. Homes, ceremonial sites, and other notable locations are often intentionally oriented based on the equinox.
In terms of navigation, knowing the solar declination angle for a specific time of year allows observers to measure the same angle down or up from where the sun rose during sunrise, thereby determining the direction of east.
To approximate the solar declination angle \(\delta\) in degrees, we can use the following equation derived in [2.1.6.1]
Graph the solar declination angle over time. Use the horizontal axis for \(N\text{,}\) the day of the year, and the vertical axis for \(\delta\text{,}\) representing the solar declination angle in degrees.
Wave heights, defined as the vertical distance between the crest and the trough of a wave, can vary in the open ocean. However, the height of the wave alone does not necessarily indicate a calm or choppy sailing conditions. Another important factor is the wave period, representing the time between waves, which affects the smoothness of sailing. For each given equation, where \(w(t)\) is the number of feet the wave is above the mean sea level at \(t\) seconds, calculate: a) the wave heigh; b) the wave period; c) Plot the wave height for two periods.
