Which of the following functions have a vertical asymptote for values of theta such that cos theta = 1? Select two answers.
y = sin theta
y = cos theta
y = tan theta
y = sec theta
y = csc theta
y = cot theta
I know what all these functions look like, but I don't know what the question means.
I seriously still have no idea, my friend suggested it's tan and cot, would this be correct?
it's cscθ and cotθ ^^
cosθ=1 at θ=0,2π,4π, ...
sinθ=0 at those same values
so, cscθ = 1/sinθ has asymptotes there
cotθ does also, but in addition it has asymptotes at the other multiples of π
The question is asking which of the given functions have a vertical asymptote for values of theta that make cos theta equal to 1.
To determine this, we need to understand what a vertical asymptote is in the context of trigonometric functions. In general, a vertical asymptote occurs when the function approaches positive or negative infinity as the input approaches a certain value.
For trigonometric functions, vertical asymptotes can occur when the function becomes undefined. In this case, the function will have a vertical asymptote when the denominator of the function is equal to zero.
Now, let's examine each function and determine if it has a vertical asymptote for cos theta = 1:
1. y = sin theta: The function y = sin theta does not have a denominator, so it does not have a vertical asymptote for any value of theta.
2. y = cos theta: The function y = cos theta does not have a denominator, so it also does not have a vertical asymptote for any value of theta.
3. y = tan theta: The tangent function is defined as y = sin theta / cos theta. This means that for values of theta where cos theta = 1, the denominator is not equal to zero. Therefore, the function y = tan theta does not have a vertical asymptote for cos theta = 1.
4. y = sec theta: The secant function is defined as y = 1 / cos theta. For values of theta such that cos theta = 1, the denominator becomes 1, and thus the function becomes undefined. As a result, the function y = sec theta has a vertical asymptote for cos theta = 1.
5. y = csc theta: The cosecant function is defined as y = 1 / sin theta. Since the denominator is sin theta, which is not equal to zero for values of theta where cos theta = 1, the function y = csc theta does not have a vertical asymptote for cos theta = 1.
6. y = cot theta: The cotangent function is defined as y = cos theta / sin theta. Similarly to the tangent function, the denominator sin theta is not equal to zero for values of theta where cos theta = 1. Therefore, the function y = cot theta does not have a vertical asymptote for cos theta = 1.
In conclusion, the two functions that have a vertical asymptote for values of theta such that cos theta = 1 are y = sec theta and y = csc theta.