h(t)=tan theta (5pie/3,3pie)
graph the function on the given interval
5 pi/3 = 300 degrees
3 pi = 540 degrees
so from -60 degrees around circle then to +180 degrees
pick some points and do it
be careful around 90 and 270 because tan is undefined along y axis
check your graph against this one:
http://www.wolframalpha.com/input/?i=tan%28x%29+for+5pi%2F3+%3C%3D+x+%3C%3D+3pi
To graph the function h(t) = tan(theta) on the interval [5π/3, 3π], we first need to understand the behavior of the tangent function.
The tangent function, tan(x), is defined as the ratio of the sine to the cosine function: tan(x) = sin(x)/cos(x). It is periodic with a period of π, meaning it repeats every π units. The tangent function also has vertical asymptotes at odd multiples of π/2, which means it shoots off to positive or negative infinity at these values.
Now, let's focus on the given interval [5π/3, 3π]. We notice that this interval spans more than one full period of the tangent function.
To graph h(t) = tan(theta) on this interval, follow these steps:
1. Determine the range of theta values within the interval.
- In this case, the range of theta values is [5π/3, 3π].
2. Calculate the corresponding tangent values for the range of theta values.
- Plug each value of theta into the tangent function: tan(theta).
- For each theta value, compute the tangent using a calculator or mathematical software.
3. Plot points on a graph.
- Set up a coordinate system with theta as the x-axis and h(t) as the y-axis.
- Plot each theta value from the interval on the x-axis and its corresponding tangent value on the y-axis.
- Connect the plotted points with a curve.
4. Identify any vertical asymptotes.
- As mentioned earlier, the tangent function has vertical asymptotes at odd multiples of π/2.
- Check if any of the theta values in the interval are odd multiples of π/2.
- If they are, draw vertical dashed lines to represent the vertical asymptotes.
By following these steps, you can graph the function h(t) = tan(theta) on the interval [5π/3, 3π].