ig. x = sin(t), y = csc(t), 0 < t < x/2 (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases y y X -1 -2 y y 6 6 5 5 4 4 3 3- 1
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Please solve the problem below. Thank you!
![### Example Problem: Parametric Equations and Curve Sketching
#### Consider the following:
\[
x = \sin(t), \quad y = \csc(t), \quad 0 < t < \frac{\pi}{2}
\]
---
**(a) Eliminate the parameter to find a Cartesian equation of the curve.**
**Solution:**
To eliminate the parameter \( t \), we start with the given parametric equations:
\[ x = \sin(t) \]
\[ y = \csc(t) \]
Recall that \(\csc(t) = \frac{1}{\sin(t)}\). Therefore, we can write:
\[ y = \frac{1}{x} \]
Hence, the Cartesian equation of the curve is:
\[ y = \frac{1}{x} \]
---
**(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter \( t \) increases.**
Below are the sketches of the curve for the given parametric equations. Each graph shows the same curve from different quadrants and views, indicating the direction with an arrow to show how the curve is traced as \( t \) increases:
1. **First Graph**
- **Axes:** The x-axis ranges from -1 to 1, the y-axis ranges from -6 to -1.
- **Curve:** The curve starts in the second quadrant and moves downward towards the third quadrant as \( t \) increases. An arrow on the curve indicates this direction.
2. **Second Graph**
- **Axes:** The x-axis ranges from -1 to 1, the y-axis ranges from -6 to -1.
- **Curve:** Similar to the first graph, this graph shows the same direction and movement of the curve in the coordinate plane as \( t \) increases.
3. **Third Graph**
- **Axes:** The x-axis ranges from 1 to -1, the y-axis ranges from 1 to 6.
- **Curve:** The curve starts from a high positive y-value on the right-side of the first quadrant and moves downward as \( t \) increases. An arrow indicates the direction of the curve.
4. **Fourth Graph**
- **Axes:** The x-axis ranges from 1 to -1, the y-axis ranges from 1 to 6.
- **Curve:** Again, the graph shows](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1148f2e-12e2-4d7a-aa19-e54736499ad8%2F9c42ee1c-6f92-4223-b29d-9fc82182aec8%2Fzyyhn1i_processed.png&w=3840&q=75)
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