Problem 102 - Algebra and Trigonometry

Question

Exercises 102–104 will help you prepare for the material covered in the next section. a. Graph \(y=\sin x\) for \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) b. Based on your graph in part (a), does \(y=\sin x\) have an inverse function if the domain is restricted to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] ?\) Explain your answer. c. Determine the angle in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) whose sine is \(-\frac{1}{2} .\) Identify this information as a point on your graph in part (a).

Step-by-Step Solution

Answer

The sine function does have an inverse in the given interval. Additionally, the angle in the interval whose sine is -1/2 is \(x = -π/6\).

Step 1: Graphing sin(x)

Plot a graph of \(y = \sin x\) between \(x = -\frac{\pi}{2}\) and \(x = \frac{\pi}{2}\). The graph starts at the origin (0, 0), rises to a peak at (π/2, 1), then descends back to the origin.

Step 2: Analyzing the function for its inverse

Analyzing the graph, we observe that for each value of \(y\) in the interval \([-1, 1]\), there is precisely one value of \(x\) in the interval \([-π/2, π/2]\). Hence, it's obtainable to conclude that the sine function has an inverse in the given interval.

Step 3: Determining the angle in given interval

We're asked to find an angle whose sine is -1/2 within the interval \([-π/2, π/2]\). The angle whose sine gives -1/2 within this interval is \(x = -\frac{π}{6}\). This can be highlighted as a point in the graph and is located at \(-π/6, -1/2 \).

Key Concepts

Graphing the Sine Function

Understanding the sine function begins with visual representation, commonly known as graphing the function. When graphing the sine function, you typically start by plotting its values over a range of angles measured in radians. The function exhibits a repeating wave-like pattern, also known as a sinusoidal curve. The graph is characterized by a smooth oscillation above and below the x-axis.

For the exercise provided, the sine function, represented as \( y = \text{sin}x \), is graphed within the interval \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \). This segment represents exactly half of a sine wave, starting and ending at the x-axis, peaking at \( \frac{\pi}{2}, 1 \) and reaching a trough at \( -\frac{\pi}{2}, -1 \).

The significance of graphing the sine function in this way allows us to observe its behavior and understand its characteristics, such as its amplitude, period, and phase shift, which are paramount to solving trigonometric problems.

Trigonometric Inverse Functions

Inverse functions are a way to 'backtrack' from the output of a function to its input. For trigonometric functions, this involves finding the angle that corresponds to a specific sine value. However, because trigonometric functions are periodic and not one-to-one, their inverses don't naturally exist over their entire domain.

In the case of the sine function, the inverse can be defined by restricting the domain. This ensures that for every sine value, there is exactly one angle that corresponds to it within this restricted domain. In our example, we determine that sine function has an inverse when the domain is limited to the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

Understanding inverse trigonometric functions is crucial because they come into play when solving equations and in applications involving right-angled triangles and waveform analysis.

Unit Circle Trigonometry

The unit circle is a fundamental tool in trigonometry, representing all the trigonometric functions geometrically. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle measured from the positive x-axis, with the coordinates of the point being \( (\cos(\theta), \sin(\theta)) \).

In unit circle trigonometry, the sine function measures the y-coordinate of a point on the unit circle. This approach can help visualize why the sine function repeats every \( 2\pi \) radians and why it has a maximum of 1 and a minimum of -1. As the angle progresses around the circle, the sine value oscillates, which reflects the wave-like pattern we see on its graph.

Sine Function Domain Restriction

To ensure the inverse sine function exists and is a function itself, you must restrict the domain of the original sine function to an interval where each y-value corresponds to one and only one x-value. In trigonometry, a common restriction for the sine function is the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), as it encompasses one complete cycle of the sine curve, from its increasing to decreasing phase, without repetition.

The exercise highlights that by restricting the domain of \( y = \sin x \) to \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), the function becomes one-to-one, making it eligible for an inverse function, written as \( y = \sin^{-1}(x) \) or \( y = \text{arcsin}(x) \). This restricted domain is particularly important when solving for angles in trigonometric equations as it simplifies the process by providing a unique solution.