Problem 2 - Precalculus Essentials

Question

Find the exact value of each expression. $$\sin ^{-1} 0$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Inverse Sine Function

The inverse sine function, \(\sin^{-1}(x)\), is the reverse operation of the sine function. It is used to find the angle that produces a specific sine value. In this case, we are asked to find the angle that produces a sine value of 0. Remembering the unit circle will be beneficial . It is also important to note that the range of \(\sin^{-1}(x)\) is \([-90^\circ, 90^\circ]\) or \([-π/2, π/2]\) in radian.

Step 2: Find the Angle

To find the inverse sine of 0, we need to identify an angle between \(-π/2\) and \(π/2\) that has a sine value of 0. On the unit circle, the sine value represents the y-coordinate. Therefore, we are looking for an angle where the y-coordinate equals 0. This occurs at 0 radians or 0 degrees.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that represents all the possible angles and their corresponding sine, cosine, and tangent values. Imagine a circle with a radius of one unit centered at the origin of a coordinate plane. The circle's edge touches the x-axis at 1 and -1, and the y-axis at 1 and -1.

When we talk about the inverse sine function, or any trigonometric function, we often refer to the unit circle to help us visualize the problem. As your exercise suggests, understanding where the sine value is zero on the unit circle is crucial. On the unit circle, the sine value corresponds to the y-coordinate of the point where the radius and the circle intersect.

When you look at the unit circle, you will notice that at 0 radians (0 degrees) and \(\pi\) radians (180 degrees), the y-coordinate is zero. However, since the range for the inverse sine function is \( [-\pi/2, \pi/2] \) or \( [-90^\circ, 90^\circ] \) in degrees, the correct angle for your problem is 0, as it’s the angle within this principal range that gives us a sine value of zero.

Radian Measure

Radian measure is a way of expressing angles based on the distance traveled around the unit circle. Unlike degrees, which divide a circle into 360 equal parts, radians are based on the circumference of the unit circle. The full circumference corresponds to \( 2\pi \) radians, making \( \pi \) radians equal to half the circle, or 180 degrees.

To convert from degrees to radians, you multiply by \( \frac{\pi}{180} \) and for the reverse, you multiply by \( \frac{180}{\pi} \). Understanding this is essential when working with trigonometric functions in calculus and physics, as they often use radian measure.

For example, when the problem requires finding the inverse sine of 0, knowing that the answer should be stated in radians is important. As you correctly determined from the unit circle, the angle whose sine is 0 is at 0 radians. Being comfortable with radians allows you to easily work within the range of the inverse sine function, which is typically stated in radians.

Sine Value

The sine value of an angle in the unit circle represents the vertical coordinate (y-coordinate) of the point where the terminal side of the angle intersects the unit circle. Specifically, it reflects how high or low the point is relative to the center of the circle.

When an angle's sine value is 0, it means that the point lies on the x-axis, either at \( (1, 0) \) or \( (-1, 0) \) on the unit circle. For angles with positive or negative sine values, the points are above or below the x-axis, respectively.

In trigonometry, it's particularly useful to remember the sine values of commonly studied angles, such as 0, \( \frac{\pi}{6} \) (30 degrees), \( \frac{\pi}{4} \) (45 degrees), \( \frac{\pi}{3} \) (60 degrees), and \( \frac{\pi}{2} \) (90 degrees). The pattern of these sine values provides a simple way to recognize or predict the sine values of other angles. Knowing that the sine of 0 is zero, as shown in your exercise, forms part of the essential groundwork for understanding trigonometry and solving more complex problems in the subject.