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Chapter 6: Problem 2

Find the exact value of the expression whenever it is defined. (a) \(\sin ^{-1}\left(-\frac{1}{2}\right)\) (b) \(\cos ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) (c) \(\tan ^{-1}(-1)\)

Short Answer

Expert verified
(a) \(-\frac{\pi}{6}\), (b) \(\frac{3\pi}{4}\), (c) \(-\frac{\pi}{4}\).

Step by step solution

01

Understand Arcsine Definition

The function \( \sin^{-1} \) is the inverse of \( \sin \), where the range is restricted to \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\). We need to find the angle \( y \) such that \( \sin(y) = -\frac{1}{2} \).
02

Determine Angle for Sin Inverse

Consider the angle \( y = -\frac{\pi}{6} \) because \( \sin(-\frac{\pi}{6}) = -\frac{1}{2} \). This satisfies the range \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\). So, \( \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \).
03

Understand Arccosine Definition

The function \( \cos^{-1} \) is the inverse of \( \cos \) with range \( 0 \leq y \leq \pi \). We need to find the angle \( y \) such that \( \cos(y) = -\frac{\sqrt{2}}{2} \).
04

Determine Angle for Cos Inverse

The angle \( y = \frac{3\pi}{4} \) satisfies \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) and fits within the range. Therefore, \( \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} \).
05

Understand Arctangent Definition

For \( \tan^{-1} \), the range is \(-\frac{\pi}{2} < y < \frac{\pi}{2}\). We need \( \tan(y) = -1 \).
06

Determine Angle for Tan Inverse

The angle \( y = -\frac{\pi}{4} \) satisfies \( \tan(-\frac{\pi}{4}) = -1 \) and is within the range. So, \( \tan^{-1}(-1) = -\frac{\pi}{4} \).

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Key Concepts

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

Arcsine
The arcsine function, denoted as \( \sin^{-1}(x) \), is the inverse of the sine function. This means it allows us to find the angle whose sine is \( x \). However, since the sine function is not one-to-one over all real numbers, we restrict its range to \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\) to define its inverse properly.

Key points about Arcsine:
  • The arcsine function returns an angle, say \( y \), for a given sine value \( x \).
  • For \( \sin^{-1}(-\frac{1}{2}) \), we determine which angle in the valid range gives a sine of \(-\frac{1}{2}\).
  • We find that \( y = -\frac{\pi}{6} \) satisfies \( \sin(y) = -\frac{1}{2} \) and falls within the specified range.
  • In terms of solution, \( \sin^{-1}(-\frac{1}{2}) = -\frac{\pi}{6} \).
When using the arcsine function, remember that the output will always be an angle in the range of \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \). This ensures that the function is well-defined and can be helpful in trigonometric equations where the sine value is known, and you need to find the angle.
Arccosine
The arccosine function, represented as \( \cos^{-1}(x) \), is the inverse of the cosine function. Unlike sine, the range for the cosine inverse is from \( 0 \) to \( \pi \) radians. This is because the cosine function is not one-to-one over all numbers, so we restrict it to this specific range.

Key points about Arccosine:
  • Arccosine gives us an angle for a given cosine value \( x \).
  • The problem asks for \( \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) \), looking for an angle where the cosine is \(-\frac{\sqrt{2}}{2}\).
  • We determine \( y = \frac{3\pi}{4} \) fits this pattern, with \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \).
  • This angle is also within the valid range of \( 0 \) to \( \pi \).
  • Thus, \( \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} \).
Understanding arccosine is essential because it frequently appears when solving problems where the angle corresponding to a cosine value is needed. By recognizing the restricted domain and range, we can accurately determine these angles.
Arctangent
Arctangent, written as \( \tan^{-1}(x) \), reverses the tangent function, returning an angle for a given tan value \( x \). The typical range for arctangent is \(-\frac{\pi}{2} < y < \frac{\pi}{2}\), allowing us to handle both positive and negative inputs.

Key points about Arctangent:
  • Arctangent helps in finding the angle \( y \) for which \( \tan(y) = x \).
  • Considering \( \tan^{-1}(-1) \), we need an angle with a tangent of \(-1\).
  • The angle \( y = -\frac{\pi}{4} \) satisfies \( \tan(-\frac{\pi}{4}) = -1 \).
  • It adheres to the range \(-\frac{\pi}{2} < y < \frac{\pi}{2}\).
  • Therefore, \( \tan^{-1}(-1) = -\frac{\pi}{4} \).
In practical applications, the arctangent function is especially useful when dealing with scenarios involving angles formed by opposite and adjacent sides, such as in right triangles or slope calculations in coordinate geometry.

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Most popular questions from this chapter

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