Problem 43 - Precalculus Essentials

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$\sin ^{-1}(\sin \pi)$$

Step-by-Step Solution

Answer

The exact value of the expression \(\sin^{-1}(\sin \pi)\) is 0.

Step 1: Find the value of sin pi

The sine of any integer number of pi is always 0. Therefore, \(\sin(\pi) = 0\).

Step 2: Perform the inverse sine operation

The inverse of sine operation of 0, also known as arcsine, is equivalent to the angle whose sine is 0. This is written as \(\sin^{-1}(0)\) and the value is 0 in the range of -pi/2 to pi/2 as per the definition of inverse sine function.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, describing the relationship between the angles and sides of right-angled triangles. It is crucial in understanding wave patterns and circular motion as well. 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 in a right triangle.
The sine function is commonly represented as \( \sin(\theta) \), where \( \theta \) is the angle in question. One of the key properties of the sine function is its periodicity. It repeats its values in regular intervals of \( 2\pi \) radians or \( 360^\circ \).
For example, \( \sin(\pi) = 0 \), because at an angle of \( \pi \) radians \((180^\circ)\), the position on the unit circle corresponds to zero on the vertical axis. This periodic nature is why \( \sin(k\pi) = 0 \) for any integer \( k \). Understanding these properties helps solve many trigonometric problems efficiently.

Arcsine

The arcsine function, also known as \( \sin^{-1}(x) \), is the inverse of the sine function. It is used to find the angle whose sine is a given number. That means if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \).
The arcsine function has a unique range typically from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians \((-90^\circ\) to \(90^\circ)\). This range is necessary to make the function one-to-one, ensuring each value maps precisely to one angle. Using this function is essential when you know the sine value and need to find the original angle.
In the exercise, when we calculate \( \sin^{-1}(0) \), we find the angle within the defined range whose sine is 0, which is simply 0 radians \((0^\circ)\). This illustrates how arcsine helps us retrieve angle values from known sine values.

Exact Trigonometric Values

Exact trigonometric values are specific values that can be determined without a calculator. These values are often derived from well-known angles in the unit circle, including \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\), and so forth.
Keeping these values in mind helps tackle trigonometric problems accurately and swiftly. For example, the sine and cosine of special angles like \(0\), \(\pi/2\), and \(\pi\) have straightforward exact values:

• \( \sin(0) = 0 \)
• \( \cos(0) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \cos(\pi) = -1 \)

Being familiar with these values is critical when evaluating expressions like \( \sin^{-1}(\sin(\pi)) \), leading to precise and reliable solutions. Memorizing these common trigonometric values simplifies the solving process of numerous problems in mathematics and physics.