Problem 77 - Calculus: Graphical, Numerical, Algebraic

Question

Controlling Outputs Let \(f ( x ) = \sin x\) (a) Find $$f ( \pi / 6 )$$ (b) Use a graph to estimate an interval \(( a , b )\) about \(x = \pi / 6\) so that $$0.3 < f ( x ) < 0.7$$ provided $$a < x < b$$ (c) Use a graph to estimate an interval \(( a , b )\) about \(x = \pi / 6\) so that $$0.49 < f ( x ) < 0.51$$ provided $$a < x < b$$

Step-by-Step Solution

Answer

We can find the solution to this exercise as : a) \(f(\pi/6) = 0.5 \) b) The interval can be \(\epsilon_1 < x-\pi/6 < \epsilon_2\) for some \(\epsilon_1, \epsilon_2 > 0\) to keep \(0.3 < f(x) < 0.7\). c) Similarly, the interval can be \(\delta_1 < x-\pi/6 < \delta_2\) for some \(\delta_1, \delta_2 > 0\) to keep \(0.49 < f(x) < 0.51\). Note that these are approximations; actual values depend on the required precision and the characteristics of the sine function.

Step 1: Find \(f(\pi/6)\)

Use the property of sine function that \(\sin(\pi/6) = 0.5\). So, \(f(\pi/6) = 0.5\)

Step 2: Estimate the Interval about \(x = \pi/6\) for \(0.3 < f(x) < 0.7\)

Look at the graph of the sine function. The function increases in the interval \((0, \pi/2)\) and reaches its maximum at \(\pi/2\). After \(\pi/2\), it decreases till \(\pi\). So we can observe the regions where \(f(x) > 0.3\) and \(f(x) < 0.7\). After estimating, we found interval about \(\epsilon_1 < x-\pi/6 < \epsilon_2\) where \(\epsilon_1, \epsilon_2 > 0\) that makes \(0.3 < f(x) < 0.7\).

Step 3: Estimate the Interval about \(x = \pi/6\) for \(0.49 < f(x) < 0.51\)

Use the graph of the sine function just like in the previous step. Only this time, the allowed discrepancy of the function from 0.5 (peak at \(\pi/6\)) is smaller: 0.01. Approximation gives us another interval about \(\delta_1 < x-\pi/6 < \delta_2\) where \(\delta_1, \delta_2 > 0\) that makes \(0.49 < f(x) < 0.51\).

Key Concepts

Solving Trigonometric Equations

Trigonometric equations are an essential part of mathematics, particularly when dealing with periodic phenomena such as waves and oscillations. Solving a trigonometric equation usually involves finding the values of variables that make the equation true. In the context of the sine function, solving an equation like \( f(x) = \sin(x) \) can often be done by recalling standard values of the sine function for specific angles.

For example, one of the most frequently used angles is \( \pi/6 \) or 30 degrees, where \( \sin(\pi/6) = 0.5 \). Knowing this property allows us to determine that for \( f(\pi/6) \) in the given exercise, the solution is simply \( f(\pi/6) = 0.5 \).

When we deal with solving more complex trigonometric equations, we often need to use methods like:

• Algebraic manipulation to isolate the trigonometric function
• Use of inverse trigonometric function to find the angle that corresponds to a given sine value
• Reference to the unit circle to understand the symmetry and periodicity of the sine function

Remember, the goal is to find all angle solutions within a given interval that satisfy the equation, which requires a deep understanding of the behavior of the sine function.

Graphing Sine Functions

Graphing is a powerful visual tool in mathematics, especially when it comes to understanding the behavior of trigonometric functions. The sine function, expressed as \( y = \sin(x) \), produces a smooth, continuous wave that oscillates above and below the x-axis. Knowing how to graph this function is crucial for predicting its behavior over specific intervals.

To graph a sine function, one must consider the following:

• The amplitude (the peak value of the wave)
• The period (how often the wave completes one full cycle)
• The phase shift (the horizontal shift of the graph)

For a basic sine function without any transformation, the period is \( 2\pi \), meaning the wave repeats every \( 2\pi \) units along the x-axis. It has no phase shift and an amplitude of 1.

In our exercise, by graphing the basic sine function, we can observe how the sine value fluctuates over time. The waves peak at \( \frac{\pi}{2} \) and trough at \( \frac{3\pi}{2} \), returning to zero at integer multiples of \( \pi \). Observing the graph near \( \pi/6 \), we can visually estimate the intervals where the sine values fall within certain boundaries, like \( 0.3 \) or \( 0.7 \), which is part of interval estimation.

Interval Estimation

Interval estimation is a critical concept in mathematics, widely used in trigonometry and calculus to approximate the ranges for solutions or function values. It involves predicting where on the number line a particular value or set of values may fall.

In trigonometry, specifically when dealing with the sine function, interval estimation helps us to determine where a function's output lies within desired bounds. In the context of the given exercise, we are interested in two specific intervals around \( x = \pi/6 \):

• An interval where \( 0.3 < f(x) < 0.7 \)
• A narrower interval where \( 0.49 < f(x) < 0.51 \)

Using a graph, we can visualize the sine curve and pinpoint where the function's output falls within these ranges. It can help us estimate the values of \( a \) and \( b \) that define these intervals such that \( a < x < b \). The precision of our estimation can vary, and the use of graphing technology can enhance the accuracy of our results.

Understanding how to estimate intervals is not only useful in solving equations but also essential in real-world applications such as signal processing, where exact values are not always necessary, and ranges are more applicable.