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Magazine Chapter 3: Problem 1
Evaluate the following: (a) \(\sin ^{-1}(0.75)\) (b) \(\cos ^{-1}(0.625)\) (c) \(\tan ^{-1} 3\) (d) \(\sin ^{-1}(-0.9)\) (e) \(\cos ^{-1}(-0.75)\) (f) \(\tan ^{-1}(-3)\)
Short Answer
Expert verified
(a) 0.848, (b) 0.895, (c) 1.249, (d) -1.119, (e) 2.418, (f) -1.249
Step by step solution
01
Understanding Inverse Sine Function
The notation \( \sin^{-1}(x) \) is used to represent the inverse sine function. It returns an angle \( \theta \) for which \( \sin(\theta) = x \). The output angle is usually measured in radians and lies within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
02
Evaluate \( \sin^{-1}(0.75) \)
To find \( \theta = \sin^{-1}(0.75) \), we look for an angle \( \theta \) such that \( \sin(\theta) = 0.75 \). Using a calculator, \( \theta \approx 0.848 \) radians.
03
Understanding Inverse Cosine Function
The notation \( \cos^{-1}(x) \) is used to represent the inverse cosine function. It returns an angle \( \theta \) for which \( \cos(\theta) = x \). The output angle is usually measured in radians and lies within the range \([0, \pi]\).
04
Evaluate \( \cos^{-1}(0.625) \)
To find \( \theta = \cos^{-1}(0.625) \), we look for an angle \( \theta \) such that \( \cos(\theta) = 0.625 \). Using a calculator, \( \theta \approx 0.895 \) radians.
05
Understanding Inverse Tangent Function
The notation \( \tan^{-1}(x) \) is used to represent the inverse tangent function. It returns an angle \( \theta \) for which \( \tan(\theta) = x \). The output angle is usually measured in radians and lies within the range \((-\frac{\pi}{2}, \frac{\pi}{2})\).
06
Evaluate \( \tan^{-1}(3) \)
To find \( \theta = \tan^{-1}(3) \), we look for an angle \( \theta \) such that \( \tan(\theta) = 3 \). Using a calculator, \( \theta \approx 1.249 \) radians.
07
Evaluate \( \sin^{-1}(-0.9) \)
For \( \theta = \sin^{-1}(-0.9) \), we need \( \sin(\theta) = -0.9 \). Using a calculator, \( \theta \approx -1.119 \) radians as the angle lies within \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
08
Evaluate \( \cos^{-1}(-0.75) \)
To find \( \theta = \cos^{-1}(-0.75) \), we need \( \cos(\theta) = -0.75 \). Using a calculator, \( \theta \approx 2.418 \) radians as the angle lies within \([0, \pi]\).
09
Evaluate \( \tan^{-1}(-3) \)
For \( \theta = \tan^{-1}(-3) \), we need \( \tan(\theta) = -3 \). Using a calculator, \( \theta \approx -1.249 \) radians as the angle lies within \((-\frac{\pi}{2}, \frac{\pi}{2})\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inverse Sine
The inverse sine function, denoted as \( \sin^{-1}(x) \) or arcsin, finds the angle whose sine is \( x \). This function answers the question: "For which angle is the sine equal to \( x \)?". The result, or angle, is usually measured in radians and restricted to the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
This means that the angle returned by the inverse sine will always be between \(-90^\circ\) and \(90^\circ\). For example:
This means that the angle returned by the inverse sine will always be between \(-90^\circ\) and \(90^\circ\). For example:
- For \( \sin^{-1}(0.75) \), we determine an angle \( \theta \) such that \( \sin(\theta) = 0.75 \). Using a calculator gives us \( \theta \approx 0.848 \) radians.
- Negative values can also be used with this function. For instance, \( \sin^{-1}(-0.9) \) means we need \( \sin(\theta) = -0.9 \), which returns an angle \( \theta \approx -1.119 \) radians.
Inverse Cosine
When you see \( \cos^{-1}(x) \), it refers to the inverse cosine function or arccos. It gives you the angle for which the cosine is \( x \). People often ask, "What angle has a cosine of \( x \)?". The angle provided by this function is measured in radians, usually within the range \([0, \pi]\), or \(0\) to \(180^\circ\).
This means the angle returned will always be between \(0\) and \(180^\circ\). Consider these examples:
This means the angle returned will always be between \(0\) and \(180^\circ\). Consider these examples:
- For \( \cos^{-1}(0.625) \), we find an angle \( \theta \) such that \( \cos(\theta) = 0.625 \). By using a calculator, we get \( \theta \approx 0.895 \) radians.
- For negative inputs, such as \( \cos^{-1}(-0.75) \), we look for \( \cos(\theta) = -0.75 \), resulting in an angle \( \theta \approx 2.418 \) radians.
Inverse Tangent
The inverse tangent function, also known as \( \tan^{-1}(x) \) or arctan, helps us find the angle for which the tangent is \( x \). Essentially, it addresses the question: "Which angle has a tangent equal to \( x \)?". Angles are typically represented in radians, with results ranging from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
The angle found will always be between \(-90^\circ\) and \(90^\circ\), regardless of the sign of \( x \). Here are some examples:
The angle found will always be between \(-90^\circ\) and \(90^\circ\), regardless of the sign of \( x \). Here are some examples:
- Evaluating \( \tan^{-1}(3) \) means finding a \( \theta \) such that \( \tan(\theta) = 3 \), and the calculator indicates \( \theta \approx 1.249 \) radians.
- For a negative value like \( \tan^{-1}(-3) \), you need \( \tan(\theta) = -3 \), giving us \( \theta \approx -1.249 \) radians.
