Problem 34
Question
On the graph of \(y=\sin x,\) points \(P\) and \(Q\) are at consecutive lowest and highest points. Find the slope of the line through \(P\) and \(Q\).
Step-by-Step Solution
Verified Answer
The slope is \(-\frac{2}{\pi}\).
1Step 1: Identify the coordinates of point P
The point P is at the lowest point on the graph of \(y = \sin x\). The lowest point of the sine graph is at \(y = -1\), and this occurs at \(x = \frac{3\pi}{2}\). Thus, the coordinates of P are \(\left(\frac{3\pi}{2}, -1\right)\).
2Step 2: Identify the coordinates of point Q
The point Q is at the highest point on the graph of \(y = \sin x\). The highest point of the sine graph is at \(y = 1\), and this occurs at \(x = \frac{\pi}{2}\). Thus, the coordinates of Q are \(\left(\frac{\pi}{2}, 1\right)\).
3Step 3: Recall the formula for the slope of a line
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
4Step 4: Calculate the slope of the line through points P and Q
Substitute the coordinates of P and Q into the slope formula. Here, \((x_1, y_1) = \left(\frac{3\pi}{2}, -1\right)\) and \((x_2, y_2) = \left(\frac{\pi}{2}, 1\right)\). The slope \(m\) is: \[ m = \frac{1 - (-1)}{\frac{\pi}{2} - \frac{3\pi}{2}} = \frac{2}{-\pi} = -\frac{2}{\pi} \]
5Step 5: Conclude the solution
The slope of the line through points P and Q on the sine graph is \(-\frac{2}{\pi}\).
Key Concepts
Sine FunctionSlope of a LineGraph Analysis
Sine Function
The sine function is a mathematical function that describes a smooth, wave-like pattern. It is defined as the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
In trigonometry, the sine function is crucial for analyzing oscillations, waves, and other periodic phenomena. The standard form of the sine function is given by: \( y = \sin x \) where \( x \) is the angle in radians.
This function has a range of [-1, 1], meaning that the sine values oscillate between these limits. The sine wave repeats its pattern every \(2\pi\) radians, which is called its period. The graph of the sine function starts at zero, rises to a maximum at \(x = \frac{\pi}{2}\), descends back to zero at \(x = \pi\), drops to a minimum at \(x = \frac{3\pi}{2}\), and returns to zero at \(x = 2\pi\).
Key points include maxima and minima:
In trigonometry, the sine function is crucial for analyzing oscillations, waves, and other periodic phenomena. The standard form of the sine function is given by: \( y = \sin x \) where \( x \) is the angle in radians.
This function has a range of [-1, 1], meaning that the sine values oscillate between these limits. The sine wave repeats its pattern every \(2\pi\) radians, which is called its period. The graph of the sine function starts at zero, rises to a maximum at \(x = \frac{\pi}{2}\), descends back to zero at \(x = \pi\), drops to a minimum at \(x = \frac{3\pi}{2}\), and returns to zero at \(x = 2\pi\).
Key points include maxima and minima:
- Maximum point at \( \left( \frac{\pi}{2}, 1 \right) \)
- Minimum point at \( \left( \frac{3\pi}{2}, -1 \right) \)
Slope of a Line
The slope of a line is a measure of its steepness, direction, and rate of change. Calculating slope in coordinate geometry involves using the change in the vertical direction (rise) over the change in the horizontal direction (run).
Mathematically, for two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated with the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula finds the tilt or angle of the line connecting the two points.
In the problem, points \(P\) and \(Q\) have coordinates \(\left(\frac{3\pi}{2}, -1\right)\) and \(\left(\frac{\pi}{2}, 1\right)\) respectively. Substituting these into the formula gives \[ m = \frac{1 - (-1)}{\frac{\pi}{2} - \frac{3\pi}{2}} = \frac{2}{-\pi} \] showing that the slope is \(-\frac{2}{\pi}\). This negative slope indicates the line descends from left to right.
Mathematically, for two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated with the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula finds the tilt or angle of the line connecting the two points.
In the problem, points \(P\) and \(Q\) have coordinates \(\left(\frac{3\pi}{2}, -1\right)\) and \(\left(\frac{\pi}{2}, 1\right)\) respectively. Substituting these into the formula gives \[ m = \frac{1 - (-1)}{\frac{\pi}{2} - \frac{3\pi}{2}} = \frac{2}{-\pi} \] showing that the slope is \(-\frac{2}{\pi}\). This negative slope indicates the line descends from left to right.
Graph Analysis
Graph analysis involves examining visual representations of data or functions to understand their behavior and key characteristics. In the context of the sine function, analyzing its graph helps identify features such as maxima, minima, and periodicity.
A thorough graph analysis involves considering:
A thorough graph analysis involves considering:
- Amplitude: The height from the midline to the peak of the wave, here it is 1 for the sine function.
- Period: The distance over which the function repeats, for \(y = \sin x\), it is \(2\pi\).
- Phase Shift: The horizontal shift left or right, not present in \(y = \sin x\) due to being a standard sine wave.
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