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Chapter 2: Problem 67

Evaluate (a) \(\sin ^{-1}(0.5)\) (b) \(\sin ^{-1}(-0.5)\) (c) \(\cos ^{-1}(0.5)\) (d) \(\cos ^{-1}(-0.5)\) (e) \(\tan ^{-1}(\sqrt{3})\) (f) \(\tan ^{-1}(-\sqrt{3})\)

Short Answer

Expert verified
(a) \(\frac{\pi}{6}\); (b) \(-\frac{\pi}{6}\); (c) \(\frac{\pi}{3}\); (d) \(\frac{2\pi}{3}\); (e) \(\frac{\pi}{3}\); (f) \(-\frac{\pi}{3}\).

Step by step solution

01

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions, such as \( \sin^{-1}(x), \cos^{-1}(x), \text{and } \tan^{-1}(x) \), find angles given their trigonometric values. They return specific principal values: \( \sin^{-1}(x) \) is in \([-\frac{\pi}{2}, \frac{\pi}{2}]\), \( \cos^{-1}(x) \) is in \([0, \pi]\), and \( \tan^{-1}(x) \) is in \((-\frac{\pi}{2}, \frac{\pi}{2})\).
02

Evaluate \(\sin^{-1}(0.5)\)

To find \(x\) where \( \sin(x) = 0.5 \), we use known angle values: \(x = \frac{\pi}{6} \) as \( \sin\left(\frac{\pi}{6}\right) = 0.5 \). Thus, \( \sin^{-1}(0.5) = \frac{\pi}{6} \).
03

Evaluate \(\sin^{-1}(-0.5)\)

For \( \sin(x) = -0.5 \), the corresponding angle in the accepted range is \(x = -\frac{\pi}{6} \) because \( \sin(-\frac{\pi}{6}) = -0.5 \). Hence, \( \sin^{-1}(-0.5) = -\frac{\pi}{6} \).
04

Evaluate \(\cos^{-1}(0.5)\)

For \( \cos(x) = 0.5 \), the principal angle is \(x = \frac{\pi}{3} \) since \( \cos\left(\frac{\pi}{3}\right) = 0.5 \). Therefore, \( \cos^{-1}(0.5) = \frac{\pi}{3} \).
05

Evaluate \(\cos^{-1}(-0.5)\)

For \( \cos(x) = -0.5 \), the angle within \([0, \pi]\) is \(x = \frac{2\pi}{3} \), since \( \cos\left(\frac{2\pi}{3}\right) = -0.5 \). Consequently, \( \cos^{-1}(-0.5) = \frac{2\pi}{3} \).
06

Evaluate \(\tan^{-1}(\sqrt{3})\)

We need \(x\) where \( \tan(x) = \sqrt{3} \). Known value is \(x = \frac{\pi}{3} \) because \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \). Thus, \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).
07

Evaluate \(\tan^{-1}(-\sqrt{3})\)

We find \(x\) where \( \tan(x) = -\sqrt{3} \). This angle is \(x = -\frac{\pi}{3} \) as \( \tan(-\frac{\pi}{3}) = -\sqrt{3} \). Therefore, \( \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \).

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Key Concepts

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

Trigonometric Identities
Trigonometric identities form the foundation for understanding complex relationships between angles and ratios. They help simplify calculations, allowing for easier evaluation of angles based on given trigonometric values. For instance, when dealing with the sine, cosine, and tangent functions and their inverses, these identities are essential tools.
  • Sine Identity: The basic identity is that the sine function represents the ratio of the opposite side to the hypotenuse in a right triangle. Known angles, like \( \sin(\frac{\pi}{6}) = 0.5 \) and \( \sin(-\frac{\pi}{6}) = -0.5 \), help derive inverse sine results.
  • Cosine Identity: Cosine is the ratio of the adjacent side over the hypotenuse. We use these identities like \( \cos(\frac{\pi}{3}) = 0.5 \) and \( \cos(\frac{2\pi}{3}) = -0.5 \) to find inverse cosine angles.
  • Tangent Identity: As the ratio of sine over cosine, it helps in finding angles such as \( \tan(\frac{\pi}{3}) = \sqrt{3} \) and \( \tan(-\frac{\pi}{3}) = -\sqrt{3} \) using known values.
These identities make it possible to evaluate inverse trigonometric functions by identifying which angle corresponds to a given trigonometric value.
Angle Evaluation
Angle evaluation involves determining the specific angle corresponding to a given trigonometric value using inverse trigonometric functions. Each inverse function is linked to evaluating angles within a defined range, called the range of principal values. This ensures that results are consistent and easily comparable across different problems.
  • Inverse Sine: It provides the angle \( x \) for which \( \sin(x) = y \). For example, \( \sin^{-1}(0.5) \) gives \( \frac{\pi}{6} \), while \( \sin^{-1}(-0.5) \) results in \( -\frac{\pi}{6} \).
  • Inverse Cosine: It returns the angle \( x \) such that \( \cos(x) = y \), such as \( \cos^{-1}(0.5) = \frac{\pi}{3} \) and \( \cos^{-1}(-0.5) = \frac{2\pi}{3} \).
  • Inverse Tangent: It identifies the angle \( x \) where \( \tan(x) = y \), like \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \) and \( \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \).
By using inverse trigonometric functions, we can find these principal angles reliably for multiple trigonometric scenarios.
Principal Values
Principal values ensure consistent and standardized answers when evaluating inverse trigonometric functions. Each inverse function has a defined principal range, which serves as a standard to avoid ambiguity in results.
  • Sine Inverse: The principal values lie between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), capturing the most commonly used angles in different quadrants.
  • Cosine Inverse: The range for principal values is between \(0\) and \(\pi\), ensuring that cosine values are evaluated with positive outcomes within a single semicircle.
  • Tangent Inverse: Principal values are limited to \((-\frac{\pi}{2}, \frac{\pi}{2})\), allowing a clear range just like the sine inverse, but without including \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) themselves.
These principal values help determine the angle uniquely for sine, cosine, and tangent inverses, thus maintaining clarity and consistency across mathematical computations.

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