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Chapter 1: Problem 210

Evaluate the functions. Give the exact value. \(\sin ^{-1}(-1)\)

Short Answer

Expert verified
\(-\frac{\pi}{2}\)

Step by step solution

01

Understanding Inverse Sine

The function \( ext{sin}^{-1}(x)\), also called arcsine, is the inverse of the sine function. It returns the angle \(\theta\) such that \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) and \(\sin(\theta) = x\).
02

Setting Up the Problem

Given \(\sin^{-1}(-1)\), we need to find the angle \(\theta\) for which \(\sin(\theta) = -1\). The angle must be within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
03

Finding the Angle

The angle within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) that satisfies \(\sin(\theta) = -1\) is \(\theta = -\frac{\pi}{2}\). At this angle, the sine value is exactly -1, which matches the given condition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arcsine
Understanding the arcsine function is pivotal when dealing with inverse trigonometric functions. The arcsine, denoted as \( \sin^{-1}(x) \) or \( \text{asin}(x) \), is the inverse function of sine. When you use the arcsine function, you are finding the angle \( \theta \) whose sine value is \( x \). This might seem a bit abstract, so think of it this way:

  • The arcsine helps us reverse the sine function, which means if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \).
  • The outcome of the arcsine function is an angle.
In the example given in the original exercise, we determined that the angle \( \theta = -\frac{\pi}{2} \) because it is the only angle within the allowable range that has a sine of -1.
Angle
Angles are fundamental in understanding inverse trigonometric functions like arcsine. When we seek the angle for which \( \sin(\theta) = x \), the arcsine function provides this angle. Every angle in trigonometry is measured in radians or degrees, but radians are often preferred in advanced mathematics. Here are some key points about angles in this context:

  • The angle resulting from \( \sin^{-1}(x) \) is limited to a specific range, ensuring a unique solution.
  • Common angles in radians include \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), and their negatives or equivalents.
Consequently, to find the exact value of \( \sin^{-1}(-1) \), we pinpoint the specific angle \( -\frac{\pi}{2} \) within the defined range of the arcsine function.
Trigonometric Range
When using trigonometric functions, it's crucial to consider their specific ranges. For inverse trigonometric functions such as arcsine, the output is always within a particular range. The range is the set of all possible outcomes (angles in this case). This ensures that, when solving for an angle, the solution is both accurate and unique. Here's what you need to know:

  • For \( \sin^{-1}(x) \), the range is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).
  • This range is crucial because it determines where the resulting angle must lie.
This range is important because without it, the solution wouldn't be unique. In our original exercise, this concept guides us to find that \( \sin^{-1}(-1) \) yields \( \theta = -\frac{\pi}{2} \), the only angle within the range where \( \sin(\theta) = -1 \).

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Most popular questions from this chapter

For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation \(y=e^{r t},\) where \(y\) is the percentage of radiocarbon still present in the material, \(t\) is the number of years passed, and \(r=-0.0001210\) is the decay rate of radiocarbon. If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. $$ f(x)=3+\ln x $$

Simplify the following trigonometric expressions. $$ \frac{\tan ^{2} x}{\sec ^{2} x}+\cos ^{2} x $$

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. $$y=4 \cos \left(2 x-\frac{\pi}{2}\right)$$

For the following exercises, write the equation in equivalent logarithmic form. $$ 9^{y}=150 $$
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