Write an expression for all of the vertical asymptotes of y=-3 csc(pi/4 *x)
Can you list step by step to show how to do it , I don't understand this problem !!! Thanks
you know that sin(pi/4 x)=0 when pi/4 x is a multiple of pi.
That is, pi/4 x = k*pi
x = k*pi * 4/pi = 4k
So, when x is a multiple of 4, sin(pi/4 x) is zero. That is where csc(pi/4 x) has vertical asymptotes.
To find the vertical asymptotes of the function y = -3 csc(pi/4 * x), we need to determine the values of x that would make the csc function undefined. The csc function is the reciprocal of the sine function, and it is undefined when the value of sine is zero.
Step 1: Set the argument of the csc function equal to the values that make the sine zero and solve for x. Recall that the general form of the sine function is sin(x) = 0 when x = nπ, where n is an integer.
In this case, we have pi/4 * x = nπ.
To solve for x, divide both sides of the equation by pi/4: x = (4n)π / pi.
Step 2: Simplify the expression: x = 4n, where n is any integer.
Step 3: Determine the values of x that make the csc function undefined. In this case, the csc function is undefined when the denominator of the csc function is equal to zero. The denominator is sin(pi/4 * x), so we need to find the values of x that make sin(pi/4 * x) = 0.
Step 4: Recall that the sine function is zero at integer multiples of π. So, we have sin(pi/4 * x) = 0 when pi/4 * x = nπ, where n is an integer.
Step 5: Solve for x by dividing both sides of the equation by pi/4: x = (4n)π / pi.
Step 6: Simplify the expression: x = 4n, where n is any integer.
Therefore, the values of x that make the csc function undefined (and thus give vertical asymptotes) are x = 4n, where n is any integer.