please help.
show step by step how to solve
h(x)=2 cot(x/2)
Identify period
identify x-intercepts
identify asymptotes
To solve the given function h(x) = 2 cot(x/2), we need to identify the period, x-intercepts, and asymptotes. Let's break down each step:
Step 1: Identify the Period
The period of the cotangent function is π, meaning one full cycle of the function occurs over an interval of π. To find the period of h(x) = 2 cot(x/2), we need to first identify the period of cot(x/2) and then multiply it by 2.
The period of cot(x/2) is calculated as 2π/(1/2) = 4π. Since h(x) = 2 cot(x/2), its period is 4π multiplied by 2, which gives us a period of 8π.
Step 2: Identify the x-intercepts
To find the x-intercepts, we set h(x) equal to zero and solve for x. In this case, we want to solve the equation 2 cot(x/2) = 0.
Since cotangent is reciprocal to the tangent function, we know that cot(x/2) will be equal to zero wherever tangent is undefined. The tangent function is undefined at values of x such that x/2 = nπ, where n is an integer.
Solving for x, we have x = 2nπ, where n is an integer. So the x-intercepts are all values of x that satisfy this equation.
Step 3: Identify the Asymptotes
The cotangent function has vertical asymptotes at values of x where cosine is equal to zero. Since h(x) = 2 cot(x/2), we need to find the values of x where cosine is equal to zero.
The cosine function is equal to zero at values of x such that x/2 = (n + 1/2)π, where n is an integer. Solving for x, we have x = (2n + 1)π, where n is an integer. So the asymptotes of h(x) are all values of x that satisfy this equation.
To summarize:
- The period of h(x) = 2 cot(x/2) is 8π.
- The x-intercepts occur at x = 2nπ, where n is an integer.
- The asymptotes occur at x = (2n + 1)π, where n is an integer.