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Magazine Chapter 6: Problem 8
Find the exact value of each expression, if it is defined. Express your answer in radians. (b) \(\cos ^{-1}\left(-\frac{1}{2}\right)\) (a) \(\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\) (c) \(\tan ^{-1}(-\sqrt{3})\)
Short Answer
Expert verified
(a) \(-\frac{\pi}{3}\), (b) \(\frac{2\pi}{3}\), (c) \(-\frac{\pi}{3}\)
Step by step solution
01
Interpret Inverse Functions
We are given inverse trigonometric functions: arcsine, arccosine, and arctangent. These functions return the angle whose trigonometric function equals the given value. The angles must fall within specific domains: - For \( \sin^{-1}(x) \), the range is \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\).- For \( \cos^{-1}(x) \), the range is \( 0 \leq y \leq \pi \).- For \( \tan^{-1}(x) \), the range is \(-\frac{\pi}{2} < y < \frac{\pi}{2}\).
02
Solve \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \)
Since the range for \( \sin^{-1} \) is \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \), we seek an angle \( y \) within this range where \( \sin(y) = -\frac{\sqrt{3}}{2} \). The angle \( y \) is \(-\frac{\pi}{3} \), because \( \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \).
03
Solve \( \cos^{-1}\left(-\frac{1}{2}\right) \)
For \( \cos^{-1} \), the range is \( 0 \) to \( \pi \). We need an angle \( y \) in this range such that \( \cos(y) = -\frac{1}{2} \). This angle is \( \frac{2\pi}{3} \), since \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \).
04
Solve \( \tan^{-1}(-\sqrt{3}) \)
Since the range for \( \tan^{-1} \) is \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \), we need the angle \( y \) within this range such that \( \tan(y) = -\sqrt{3} \). The angle \( y \) is \(-\frac{\pi}{3} \) because \( \tan\left(-\frac{\pi}{3}\right) = -\sqrt{3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inverse Sine
Inverse sine, often denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), is a fundamental concept in trigonometry. It refers to the angle whose sine is \( x \). The sine function, \( \sin(y) = x \), can be reversed using inverse sine to find \( y \). It is crucial to remember that the output of the inverse sine falls within a specific range.
The range of \( \sin^{-1}(x) \) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which corresponds to angles in the fourth and first quadrants. This range is defined because the sine function has only one-to-one correspondence within these bounds. Here's how it works in practice:
The range of \( \sin^{-1}(x) \) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which corresponds to angles in the fourth and first quadrants. This range is defined because the sine function has only one-to-one correspondence within these bounds. Here's how it works in practice:
- To find \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \), we determine the angle \( y \) such that \( \sin(y) = -\frac{\sqrt{3}}{2} \) within this range.
- The correct angle is \( y = -\frac{\pi}{3} \), as our given value satisfies the sine of this angle.
Inverse Cosine
Inverse cosine, denoted \( \cos^{-1}(x) \) or \( \arccos(x) \), is the angle whose cosine is \( x \). The cosine function, \( \cos(y) = x \), can be reversed with the inverse cosine function. This function has its range between \(0\) and \(\pi\) radians.
This range covers angles in the first and second quadrants, where the cosine function remains one-to-one. This ensures a unique angle is returned for every cosine value within the interval of \([-1, 1]\). In our specific case:
This range covers angles in the first and second quadrants, where the cosine function remains one-to-one. This ensures a unique angle is returned for every cosine value within the interval of \([-1, 1]\). In our specific case:
- For the expression \( \cos^{-1}\left(-\frac{1}{2}\right) \), we look for an angle \( y \) where \( \cos(y) = -\frac{1}{2} \).
- The angle \( y \) that satisfies this condition is \( \frac{2\pi}{3} \).
Inverse Tangent
Inverse tangent, labeled \( \tan^{-1}(x) \) or \( \arctan(x) \), refers to the angle with a given tangent of \( x \). It helps to deduce \( y \) when \( \tan(y) = x \). Like the other inverses, \( \tan^{-1}(x) \) has a specified range for its output.
This range is from \(-\frac{\pi}{2}\) to \( \frac{\pi}{2}\). This selection covers angles where the tangent function is continuous and uniquely defined. The domain includes the entire real line \((-\infty, \infty)\) because the tangent function tends toward infinity at odd multiples of \( \frac{\pi}{2} \).
This range is from \(-\frac{\pi}{2}\) to \( \frac{\pi}{2}\). This selection covers angles where the tangent function is continuous and uniquely defined. The domain includes the entire real line \((-\infty, \infty)\) because the tangent function tends toward infinity at odd multiples of \( \frac{\pi}{2} \).
- When solving \( \tan^{-1}(-\sqrt{3}) \), you are identifying the angle \( y \) for which \( \tan(y) = -\sqrt{3} \).
- The solution falls at \( y = -\frac{\pi}{3} \), as it meets the range and conditions of the problem.
Trigonometric Function Ranges
Each inverse trigonometric function has its own range to return unique and specific angle solutions. It is crucial to understand these ranges to correctly solve problems involving these functions.
Here's a quick overview of the specific trigonometric function ranges:
Here's a quick overview of the specific trigonometric function ranges:
- **Inverse Sine Range:** \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\), capturing angles in the fourth and first quadrants.
- **Inverse Cosine Range:** \(0 \leq y \leq \pi\), including angles from the first and second quadrants.
- **Inverse Tangent Range:** \(-\frac{\pi}{2} < y < \frac{\pi}{2}\), restricted to angles mainly in the fourth and first quadrants without including the extremes.
- These ranges ensure that each inverse function can correctly identify angles for specific trigonometric values in a consistent and understood format. This understanding assists in solving trigonometry problems with precision and ease.
