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Here are a few recommended readings before getting started with this lesson.
On the weekend, Kriz and their family headed to the local zoo. Kriz really loves learning about animals, so they were sure to stop at all the cool information boards that teach interesting facts about them.
Of all the animals, Kriz likes foxes and rabbits the most. They were dying to learn more about them and discovered a table showing the state population of rabbits and foxes during the previous year.chasethe other. What can be a possible explanation for this?
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.
Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The sine function, denoted as sin, can be defined as the y-coordinate of the point P.
Note that for x in the interval [-2π,0] and in the interval [0,2π] the graph looks exactly the same. This means that the sine function is a periodic function and its period is 2π.
sin(θ+2πn)=sinθ
Here, n is any integer number. Consider the function y=asinbθ, where a and b are non-zero real numbers and θ is measured in radians. With this information, the properties of the sine function can be defined.
Properties of y=asinbθ | ||
---|---|---|
Amplitude | ∣a∣ | |
Number of cycles in [0,2π] | ∣b∣ | |
Period | ∣b∣2π | |
Domain | All real numbers | |
Range | [-∣a∣,∣a∣] |
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
Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The cosine function, denoted as cos, can be defined as the x-coordinate of the point P.
Note that for x in the interval [-2π,0] and in the interval [0,2π] the graph looks exactly the same. This means that the cosine function is a periodic function and its period is 2π.
cos(θ+2πn)=cosθ
Here, n is any integer number. Consider the function y=acosbθ, where a and b are non-zero real numbers and θ is measured in radians. With this information, the properties of the cosine function can be defined.
Properties of y=acosbθ | ||
---|---|---|
Amplitude | ∣a∣ | |
Number of cycles in [0,2π] | ∣b∣ | |
Period | ∣b∣2π | |
Domain | All real numbers | |
Range | [-∣a∣,∣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
Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The tangent function, denoted as tan, can be defined as the ra