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Here are a few recommended readings before getting started with this lesson.
Trigonometric Function | Domain | Range |
---|---|---|
y=sinx | All real numbers | [-1,1] |
y=cosx | All real numbers | [-1,1] |
The tangent, cotangent, secant, and cosecant functions are defined as rational functions that involve the sine and cosine functions. The domain of each function does not include values that would make their denominator zero.
Trigonometric Function | Ratio | Domain | Range |
---|---|---|---|
y=tanx | tanx=cosxsinx | Real numbers except odd multiples of 2π | All real numbers |
y=cotx | cotx=sinxcosx | Real numbers except multiples of π | All real numbers |
y=secx | secx=cosx1 | Real numbers except odd multiples of 2π | (-∞,-1]∪[1,∞) |
y=cscx | cscx=sinx1 | Real numbers except multiples of π | (-∞,-1]∪[1,∞) |
Now, one of the main trigonometric functions, the sine function, will be defined and examined more closely.
In the general form y=asinbθ, the amplitude is ∣a∣ and the period is ∣b∣2π.
Properties of y=asinbθ | ||
---|---|---|
Amplitude | ∣a∣ | |
Number of cycles in [0,2π] | ∣b∣ | |
Period | ∣b∣2π |
b=2π
∣∣∣∣2π∣∣∣∣=2π
b/ca=ba⋅c
ba=b/πa/π
1a=a
Kriz visited the port on a day when a special exhibition was taking place where scientists explained how they use a submarine in ocean exploration. They learned that radars are used to monitor objects under the sea. Even more interesting, in operating radars, sine and cosine functions are involved.
A wave signal received by a radar can be modeled by the following equation.In the general form y=acosbθ, the amplitude is ∣a∣ and the period is ∣b∣2π.
Properties of y=acosbθ | ||
---|---|---|
Amplitude | ∣a∣ | |
Number of cycles in [0,2π] | ∣b∣ | |
Period | ∣b∣2π |
The midline of the parent cosine function is y=0. However, since the considered function is translated 1 unit upward, its midline is also translated. This means that the equation of the midline is y=1.
By connecting the plotted points with a smooth curve and continuing it periodically in both directions, the graph of the function can finally be drawn.
There are formulas for the key points such as x-intercepts, maximum value, and minimum value of a sine function of the form y=asinbx.
Formula | ||
---|---|---|
x-intercepts | (0,0), (21⋅b2π,0), (b2π,0) | |
Maximum | a>0 (41⋅b2π,a) |
a<0 (43⋅b2π,-a) |
Minimum |
a>0 |
a<0 (41⋅b2π,a) |
Similarly, there are also formulas for the x-intercepts, maximum, and minimum of a cosine function of the form y=acosbx.
Formula | ||
---|---|---|
x-intercepts | (41⋅b2π,0), (43⋅b2π,0) | |
Maximum | a>0 (0,a), (b2π,a) |
a<0 (21⋅b2π,-a) |
Minimum | a>0 (21⋅b2π,-a) |
a<0 (0,a), (b2π,a) |
These formulas can be useful when graphing a sine or a cosine function. By using them, the first five points of a function can be plotted. Then, the function can be extended along the x-axis by imitating the found pattern.
After learning how trigonometric functions are abundant in objects related to the ocean, Kriz was stoked to go to Physics class first thing Monday. There, they learned that light travels in waves and, therefore, can be modeled by sine and cosine functions. Different colors have different wavelengths, or periods, and the amplitude of the wave affects the brightness of the color.
For example, the light visible as red has the longest period, while the light visible as violet has the shortest period. Additionally, the greater the amplitude of the light wave, the brighter it looks.
Next, some key points, like maximums, minimums, and intersections with the midline should be plotted. The parent sine function intersects the midline at each half-period.
In this case, the period is 660, so its half-period is 2660=330 nanometers. Therefore, the x-coordinates of the intersections of the function and the midline occur at x-values that are multiples of 330.The maximums and minimums of a sine function occur once every period between two points of intersection with the midline. Analyzing the graph of the parent sine function starting from the origin, it can be seen that the maximum of the function occurs before the minimum.
Therefore, the maximum of the given function is in the middle between the intersections (0,0.5) and (330,0.5), while the minimum is in the middle between (330,0.5) and (660,0.5). By adding and subtracting the amplitude of 0.3 to the midline, the y-coordinates of the maximum and minimum, respectively, can be found.Finally, connect the points with a smooth curve and continue it periodically.
b=200π
∣∣∣∣200π∣∣∣∣=200π
b/ca=ba⋅c
ba=b/πa/π
1a=a
Next, the key points should be identified and plotted. Consider the parent cosine function.
As can be seen, the maximums occur at x=-2π,0,2π,…, which are the multiples of its period 2π. This means that using the period of 400 of the considered function, its maximums can be found.By connecting the points with a smooth curve and continuing it periodically, the graph of the given function can be obtained.
The frequency of a periodic function is the number of cycles in a given unit of time. The frequency of a function's graph is the reciprocal of the function's period.
Frequency=Period1
hertz.For instance, 10 Hz means 10 times per second.
Later that day, Kriz was excitingly sharing their impressions with their classmate Zain about their visit to the zoo. Kriz told Zain that they were impressed to learn that elephants can hear frequencies 20 times lower than humans, while mice can hear astronomically high frequencies, up to 70 - 80 kHz.
Frequency=10
LHS⋅Period=RHS⋅Period
LHS/10=RHS/10
hearwhen communicating miles apart!
Frequency=75000
LHS⋅Period=RHS⋅Period
LHS/75000=RHS/75000
Period=750001
Cross multiply
Multiply
b=2π
∣∣∣∣2π∣∣∣∣=2π
b/ca=ba⋅c
Calculate quotient
b=2π
∣∣∣∣2π∣∣∣∣=2π
2⋅2a=a
aa=1
Next, divide the period of the function into four equal parts and locate three points between the asymptotes. In this case, the period is 2, so each fourth is 0.5 units long.
Therefore, the x-coordinates of the points that will be plotted are -21, 0, and 21. Substitute these values for x into the function rule and evaluate the corresponding y-coordinates.
y=1.5tan2πx | |||
---|---|---|---|
x-Coordinate | Substitute | Simplify | Evaluate |
x=-21 | y=1.5tan2π(-21) | y=1.5tan(-4π) | y=-1.5 |
x=0 | y=1.5tan2π(0) | y=1.5tan0 | y=0 |
x=-21 | y=1.5tan2π(21) | y=1.5tan4π | y=1.5 |
Now that both coordinates of the three points are known, plot them on the coordinate plane with the asymptotes.
Finally, connect the three points with a smooth curve and continue the function to the left and right keeping in mind that it should get closer to the asymptotes but will never intersect them.
Replicate the branch to obtain the graph for other intervals.
When drawing one period of a function of the form y=atanbθ, the following characteristics of a tangent function can be used.
Formula | ||
---|---|---|
x-intercept | (0,0) | |
Asymptotes | θ=-2∣b∣π, θ=2∣b∣π | |
Halfway Points | (-4bπ,-a), (4bπ,a) |
Because the tangent function has cosine in its denominator, the asymptotes of the tangent function are located at the zeros of the cosine function.
Since the sine function is the numerator of the tangent function, the zeros of the sine function will be the zeros of the tangent function as well.
The y-values of the sine and cosine functions are equal at the intersection of both graphs. This means that the y-value of the tangent function will be 1 at the x-coordinates of these points.
Likewise, three more points can be plotted between the left asymptotes of each period and the zeros. This time, however, since the sine and cosine functions have opposite x-values, the y-value of the tangent function will be -1.
Finally, the branches are drawn by starting from the bottom of the left asymptote and moving towards the top of the right asymptote.After school, Kriz and Zain will meet some friends to play volleyball near Zain's home. Zain lives in a modern 300-feet building. As Kriz was approaching the building about 175 feet from its base, they saw Zain going down in the elevator and waved to them.
LHS⋅175=RHS⋅175
LHS+d=RHS+d
LHS−175tanx=RHS−175tanx
y=-175tanx | |||
---|---|---|---|
x-Coordinate | Substitute | Evaluate | |
x=-4π | y=-175tan(-4π) | y=175 | |
x=0 | y=-175tan0 | y=0 | |
x=4π | y=-175tan(4π) | y=-175 |
Now, plot the points on the coordinate plane with the asymptotes.
Finally, the graph of y can be drawn by connecting the points with a smooth curve such that, as x tends to -2π and 2π, the graph gets closer and closer to the asymptotes but never crosses them.
The last step is to translate the graph 300 units up vertically to obtain the graph of d.
Therefore, only the part of the graph in the first quadrant makes sense considering the context.
Earlier, it was mentioned that on the weekend Kriz went with their family to the local zoo.
Graph of f(m):
Explanation: A predator-prey relationship between foxes and rabbits.
Both functions have a general shape of a sine or a cosine function.
Period=12
LHS⋅∣b∣=RHS⋅∣b∣
LHS/12=RHS/12
ba=b/2a/2
Similarly, by connecting the points given for the function f(m), its graph can be drawn.
The fox population seems to be behind the rabbit population by three months. In other words, the fox population chases the rabbit population. This can be explained by the predator-prey relationship that exists between foxes and rabbits, as rabbits are a source of food for foxes. Where there is a big population of rabbits, the fox population grows.