sketch the graph from each function in the interval from 0 to 2π. 20=3sinØ
To sketch the graph of 20 = 3 sin Ø in the interval from 0 to 2π, we can start by isolating sin Ø:
20 = 3 sin Ø
sin Ø = 20/3
Since sin Ø is a ratio of the opposite side to the hypotenuse in a right triangle, we can interpret this as the y-coordinate of a point on the unit circle. We can find the corresponding angle Ø by using the inverse sine function:
Ø = sin⁻¹ (20/3) ≈ 1.22
This means that the point (cos Ø, sin Ø) on the unit circle that corresponds to 20/3 on the y-axis is located at an angle of approximately 1.22 radians. We can plot this point on the unit circle:
Next, we can sketch the graph of y = 3 sin Ø by multiplying the y-coordinate of each point on the unit circle by 3. This gives us:
We can see that the graph is symmetric about the x-axis and has an amplitude of 3. It reaches a maximum value of 3 at Ø = π/2 and a minimum value of -3 at Ø = 3π/2. We can use these key points to sketch the graph:
The final graph of 20 = 3 sin Ø in the interval from 0 to 2π looks like:
(Note: The dotted lines represent the x-axis and the y-axis.)
find an example of the graph you described
Here's an example of a graph that matches the description I gave for the function 20 = 3sin(Ø):
Graph of y = 3sin(Ø) with amplitude of 3, maximum value of 3 at Ø = π/2, and minimum value of -3 at Ø = 3π/2.
To sketch the graph of the function 20 = 3sin(Θ) in the interval from 0 to 2π, we can follow these steps:
Step 1: Determine the amplitude:
The coefficient of the sine function is 3, therefore the amplitude is |3| = 3.
Step 2: Determine the period:
The period of the sine function is given by 2π. Since there is no coefficient affecting the angle, the period remains the same.
Step 3: Identify the key points:
The key points on the graph of the sine function are the maximum points (amplitude) and the zero-crossings.
a) Maximum/Minimum point:
The maximum value of the function is equal to the amplitude, which is 3. So, there is a maximum point at (0, 3) and another one at (π, 3).
b) Zero-crossings:
To find the zero-crossings, we solve the equation 3sin(Θ) = 0
This gives us sin(Θ) = 0
Θ can be 0, π, 2π, etc.
Therefore, there are zero-crossings at (0, 0), (π, 0), and (2π, 0).
Step 4: Sketch the graph:
Based on the above information, we can sketch the graph as follows:
- Start at the point (0, 3) above the x-axis.
- Connect the maximum point with the zero-crossing at (π, 0).
- Continue to the next zero-crossing at (2π, 0).
- Connect the zero-crossing at (2π, 0) with the zero-crossing at (0, 0).
- Finally, connect the zero-crossing at (0, 0) with the maximum point at (0, 3).
The resulting graph will be a sine wave with an amplitude of 3, going through the points (0, 3), (π, 0), and (2π, 0), as described above.
Note: The wave will repeat itself every 2π units due to the nature of the sine function.