Question

Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. $$\sin ^{-1}(-1)$$

Short answer

Answer: The value of the expression \(\sin^{-1}(-1)\) is \(-\frac{\pi}{2}\).

Step-by-step solution

Recall inverse sine function domain and range

The inverse sine function, written as \(\sin^{-1}(x)\) or \(\arcsin(x)\), is defined only for \(x\) values in the interval \([-1, 1]\). Moreover, its range is within the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This means that for any value of \(x\in[-1, 1]\), the corresponding \(\sin^{-1}(x)\) value will lie within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Determine if the expression is within the domain

We are given the expression \(\sin^{-1}(-1)\). Here, \(x=-1\). Since \(x\) is within the interval \([-1, 1]\), the inverse sine function is defined for this value.

Evaluate the expression

Now that we know the expression is defined, we need to find the angle whose sine is equal to \(-1\). Recalling our knowledge of the unit circle, we know that the sine of an angle represents the \(y\)-coordinate on the unit circle. At an angle of \(\phi = -\frac{\pi}{2}\), the coordinate is \((0, -1)\), which means $$\sin(-\frac{\pi}{2}) = -1.$$ Therefore, we have $$\sin^{-1}(-1) = -\frac{\pi}{2}.$$

Final Answer

The value of the given expression is $$\sin^{-1}(-1) = -\frac{\pi}{2}.$$

Key concepts

Inverse Sine

The inverse sine function is a fascinating aspect of trigonometry. It's symbolized as \( \sin^{-1}(x) \) or \( \arcsin(x) \), and it allows us to determine an angle from a given sine value. This function is specifically designed to reverse the process of the sine function. Given that the sine function is not one-to-one, we restrict its domain to \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) to make it invertible. This ensures that each input in this domain has a unique output. For inverse sine, it means finding the angle within this interval whose sine value is a particular number. This function is crucial in various fields such as mathematics, engineering, and physics, as it transforms numerical values into geometric representations, aiding in problem-solving.

Unit Circle

The unit circle is a central tool in trigonometry, helping us visualize the relationships between angles and the coordinates representing sine and cosine values. Imagine a circle with a radius of 1 unit, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle measured from the positive x-axis, with the x-coordinate representing the cosine of that angle and the y-coordinate the sine. For instance:

• At \( \theta = 0 \), the point is \( (1, 0) \)
• At \( \theta = \frac{\pi}{2} \), the point is \( (0, 1) \)
• At \( \theta = \pi \), the point is \( (-1, 0) \)
• At \( \theta = -\frac{\pi}{2} \), the point is \( (0, -1) \)

This coordinated geometric view provides a powerful way to verify trigonometric identities and understand the properties of functions such as inverse sine. With the unit circle, we can trace the path of an angle and find specific sine and cosine values intuitively.

Trigonometric Function Domain and Range

Understanding the domain and range of trigonometric functions is vital for working with inverses. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For the sine function, its standard domain for becoming invertible is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), meaning it spans from minus half of pi to plus half of pi. Within this interval, the sine function covers all sines from -1 to 1, forming its range. When we look at the inverse sine \( \sin^{-1}(x) \), the roles are interchanged. Its domain becomes \([-1, 1]\), reflecting the possible output values of the standard sine function, and the range becomes \([-\frac{\pi}{2}, \frac{\pi}{2}]\), showing the angles that result in output values of \(x\). These restrictions ensure that each value yields unique results, thus allowing us to "reverse" the sine process efficiently using arcsine.

Arc Sine

Arc sine, or \( \arcsin \), is another name for the inverse sine function. Just like \( \sin^{-1} \), it represents the angle that gives rise to a specific sine value. This function is particularly useful because:

• It helps solve equations involving sine values when you need to find exact angles.
• It converts direct sine results back into angular measures, enhancing comprehension of angles within real-world problems.

An important behavior of the arc sine function is its bounded range. This means that the angles it produces will always fall within a specific interval, \([-\frac{\pi}{2}, \frac{\pi}{2}]\), providing consistency and predictability in its outputs.Understanding \( \arcsin \) helps create a bridge between linear numerical values and cyclic angular relationships, which are important in disciplines like physics and engineering.