Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, find the exact value without using a calculator. $$ \sin \left(\sin ^{-1} 0.4567\right) $$

Step-by-Step Solution

Answer

The exact value is 0.4567.

Step 1: Understand the Inverse Function

The function \(\sin^{-1}(x)\) denotes the inverse sine function, also known as arcsin, which returns the angle whose sine value is \(x\).

Step 2: Apply the Inverse Identity

By definition of the inverse function, \(\sin(\sin^{-1}(x)) = x\) for any value within the range of the inverse sine, \([-1, 1]\). Since \(0.4567\) is within this range, this identity holds true.

Step 3: Solution Conclusion

Given the calculation from a correct inverse identity application: \[\sin(\sin^{-1}(0.4567)) = 0.4567.\] This is the exact value for this equation.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the ratios of sides in right-angled triangles are known. These functions essentially reverse what the standard trigonometric functions do. While functions like sine, cosine, and tangent take an angle and give a ratio, the inverse functions take a ratio and give an angle.
The most common inverse functions are:

• Arcsine (\( \sin^{-1}(x) \)
• Arccosine (\( \cos^{-1}(x) \)
• Arctangent (\( \tan^{-1}(x) \)

These are denoted by \( \sin^{-1}, \cos^{-1}, \) and \( \tan^{-1} \) respectively, but read as "arcsin," "arccos," and "arctan." They return values in specific ranges to maintain function properties.
Inverse trigonometric functions are essential in fields like geometry and physics, where determining unknown angles is often necessary.

Sine Function

The sine function is one of the primary trigonometric functions, originating from the study of right-angled triangles. In these triangles, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. \[ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \]In the unit circle, the sine of an angle corresponds to the y-coordinate of a point on the circle.
This function repeats its values in a periodic manner, with a period of \( 2\pi \, (360^{\circ})\) for radians and degrees respectively. The range of sine is \([-1, 1]\), which indicates it can only result in these values.
Additionally, the function is often employed in physics and engineering, describing phenomena like waves and oscillations.

Arcsin

Arcsin, or the inverse sine function, is used when you need to find an angle whose sine is a given number. It is expressed as \( \sin^{-1}(x) \) or "arcsin of \( x \)."
The fundamental function of arcsin is to return the unique angle \( \theta \) in radians (or degrees), such that \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \) radians or \( -90^{\circ} \leq \theta \leq 90^{\circ} \). This limitation ensures it has a unique value for each input in its domain. \[ \sin(\sin^{-1}(x)) = x \]
A common use of arcsin is in trigonometry problems to recover the angle provided by a given sine value, perfect for applications in study fields involving waveforms, acoustics, and electrical engineering.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate trigonometric functions. They are essential for simplifying expressions and solving complex trigonometric equations. These identities help in proving or finding relationships among angles.
Some basic yet powerful trigonometric identities include:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal Identity: \( \sin(\theta) = \frac{1}{\csc(\theta)} \)
• Angle Sum and Difference Identities, like \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \)

These identities play a vital role not only in solving mathematical problems but also in practical applications like signal processing and optics, where wave properties need to be understood and manipulated.