What graph is traced by the equation (x, y) = (5sin 12t , 6−5 cos 12t)?Think of another equation that will produce the same graph. Use your calculator to check.
then
x = 5sin 12t
x/5 = sin 12t
12t = arcsin (x/5)
y = 6-5cos 12t
5cos 12t = 6-y
cos 12t = (6-y)/5
12t = arcsin (6-y)/5
construct a right angled triangle with hypotenuse 5, opposite as x, and adjacent at 6-y with a base angle of 12t
all the above would be true according to that triangle.
Also
x^2 + (6-y)^2 = 25
x^2 + 36 - 12y + y^2 = 25
x^2 + y^2 - 12y = -11
looks like the equation of a circle to me.
Oh, the graph traced by the equation (x, y) = (5sin 12t , 6−5 cos 12t) is definitely the "Crazy Coaster" graph. It's like a rollercoaster for math nerds!
Now, let me put on my hilarious hat and think of another equation that will produce the same graph. How about (x, y) = (6tan 12t , 5sec 12t)? It's like an "Extreme Math Tornado" swirling on the graph! Make sure to fasten your math seatbelts!
But don't take my word for it. Whip out that trusty calculator and see the glorious synchronization of these equations for yourself. Let the math hilarity begin!
The equation (x, y) = (5sin 12t , 6−5 cos 12t) represents a parametric equation in terms of time variable t. It traces the path of a point (x, y) on a graph over a certain time interval.
To think of another equation that will produce the same graph, we can use a trigonometric identity that relates sine and cosine functions:
cos(α) = sin(α + π/2)
By substituting this identity into the original equation, we can rewrite it as:
(x, y) = (5sin 12t , 6−5 sin(12t + π/2))
This new equation will produce the same graph as the original equation.
To verify this using a calculator, you can input some values of t into both equations and compare the resulting (x, y) coordinates.
The equation (x, y) = (5sin 12t, 6−5cos 12t) represents parametric equations, where t is the parameter and (x, y) are the coordinates of a point on the graph at a given value of t.
Let's analyze the equation to understand the shape of the graph. The x-coordinate, 5sin 12t, represents the horizontal position of the point, while the y-coordinate, 6−5cos 12t, represents the vertical position of the point.
The sine function, sin 12t, oscillates between -1 and 1, modifying the amplitude and position of the graph along the x-axis. The coefficient of 5 determines the amplitude of oscillation. Thus, the x-coordinate of each point varies between -5 and 5, resulting in horizontal oscillations.
Similarly, the cosine function, cos 12t, oscillates between -1 and 1, modifying the amplitude and position of the graph along the y-axis. The coefficient of 5 determines the amplitude of oscillation. However, since it is subtracted from 6, the graph is shifted upward by 6 units. Therefore, the y-coordinate of each point varies between 1 and 11, resulting in vertical oscillations.
Combining these oscillations in the x and y directions results in a graph shaped like an ellipse. The major axis is along the y-axis, ranging from 1 to 11, and the minor axis is along the x-axis, ranging from -5 to 5.
To find another equation that produces the same graph, we need to find equivalent parametric equations that yield the same shape. One way to achieve this is by changing the coefficients that determine the amplitude and position of the graph.
Consider the equation (x, y) = (8sin 12t, 10−8cos 12t). Here, the coefficient of 8 in the x-coordinate increases the amplitude of oscillation along the x-axis, and the coefficient of 8 in the y-coordinate increases the amplitude of oscillation along the y-axis. Adjusting the constant terms in each coordinate determines the position of the graph.
Using a calculator or plotting software, graph both equations: (x, y) = (5sin 12t, 6−5cos 12t) and (x, y) = (8sin 12t, 10−8cos 12t). By plotting these equations, you will find that they produce identical ellipses, confirming that they represent the same graph.