What are the graphs of y = cos x and y = sec x in the interval from -2pi to 2pi?
You should be familiar with the cosine curve
The cosine curve runs from -1 to +1.
cos(0) = 1
cos(60°) = 1/2
cos(90) = 0
cos(120) = -1/2
etc
since secx = 1/cosx
at any given x , the secant is the reciprocal of the cosine value
e.g. cos(60) = 1/2, thus sec(60) = 2
which means that when the cosine value is zero, (as it crosses the x-axis) , we would have the reciprocal of zero , which of course is undefined
So we have vertical asymptotes whenever the cosine value is 0.
The result is a series of U shaped curves sitting on top or falling below the cosine curve as you can seen in the graph I gave you
omg its stewie omg where's Louis
yes. that is right. and good morning mrs. sue
my mind has been shattered by these revelations
Oh, graphs, my favorite! Well, let's analyze these two hilarious characters - y = cos x and y = sec x.
First up, the cos x graph is like a super cool roller coaster ride that loops around the x-axis. It starts at the top with a hilarious peak at (0, 1), then quickly dives down to a hilarious trough at (π, -1), and then it goes back up to another peak at (2π, 1). So, picture a roller coaster with lots of ups and downs, but instead of screams, it generates lots of laughs!
Now, as for the sec x graph, hold onto your funny bones because this one is a real knee-slapper. It's like a wacky, wild ride that never stops, never ends! It starts off at negative infinity, haha! Then, it zooms up to positive infinity at (0, 1), cracks a joke at (π, -1), and then goes back down to negative infinity at (2π, 1). So, it's like a roller coaster that goes up, up, and away without ever coming down!
In summary, their graphs are like a comedy show with one roller coaster going up and down, while the other keeps skyrocketing to infinity, leaving you in an eternal state of laughter and disbelief! Enjoy the ride! 🎢🤡
To determine the graphs of y = cos x and y = sec x in the interval from -2π to 2π, it is helpful to understand the basic shapes and characteristics of these functions.
1. y = cos x:
The function y = cos x represents the cosine function. The cosine function is periodic with a period of 2π, meaning it repeats its pattern every 2π units. It oscillates between the values of -1 and 1 along the y-axis. To obtain the graph in the given interval, you can divide the interval -2π to 2π into increments of the period, which is 2π.
Start by plotting the values of x at multiples of π/2, π, and 3π/2, which are critical points for cos x. These values are -2π, -3π/2, -π, -π/2, 0, π/2, π, 3π/2, and 2π. Evaluate the corresponding y-values by substituting those x-values into the equation y = cos x. Connect the points smoothly, and you'll obtain a graph that resembles a wave-like pattern with symmetrical peaks and troughs.
2. y = sec x:
The function y = sec x represents the secant function. The secant function is the reciprocal of the cosine function, so it is defined as 1/cos x. The values of sec x are undefined when the cosine function equals zero.
To draw the graph of y = sec x, you can plot the values of x at multiples of π/2, where cos x equals zero. These values are -π/2, π/2, -3π/2, and so on. Evaluate the corresponding y-values by substituting those x-values into the equation y = sec x. To complete the graph, you can identify additional points by finding the reciprocal of the y-values obtained from the cosine graph. Connect the points smoothly, and the resulting graph will have a wave-like pattern, with vertical asymptotes where it approaches infinity as it approaches zero on the x-axis.
By following these steps, you can plot the graphs of y = cos x and y = sec x in the interval from -2π to 2π.