Question

Without using a calculator, evaluate, if possible, the following expressions. $$\sin ^{-1} \frac{\sqrt{3}}{2}$$

Short answer

Answer: \(\frac{\pi}{3}\)

Step-by-step solution

Recall the properties of the inverse sine function

The inverse sine function, also known as the arcsine function, is denoted by \(\sin^{-1}(x)\) and is defined as the angle whose sine value is equal to x. It has a domain of \(-1\leq x\leq1\) and a range of \(-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\).

Recognize the special angle

We are given the expression: $$\sin^{-1}\frac{\sqrt{3}}{2}.$$ We need to determine the angle \(\theta\) such that its sine value is equal to \(\frac{\sqrt{3}}{2}\). This can be done by remembering the values of the sine function for special angles. In the unit circle, we know that the sine values of \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) are \(\frac{1}{2}\), \(\frac{\sqrt{2}}{2}\), and \(\frac{\sqrt{3}}{2}\), respectively.

Find the value of the inverse sine function

Since we are looking for the angle whose sine value is equal to \(\frac{\sqrt{3}}{2}\), and we know that the sine of \(\frac{\pi}{3}\) is equal to \(\frac{\sqrt{3}}{2}\), we can conclude that: $$\sin^{-1}\frac{\sqrt{3}}{2} = \frac{\pi}{3}.$$ So, the value of the given expression is \(\frac{\pi}{3}\).

Key concepts

Unit Circle

The unit circle is an essential tool in trigonometry, representing all angles and their corresponding sine and cosine values. It's a circle with a radius of 1, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle and can be described using coordinates \((\cos \theta, \sin \theta)\). The x-coordinate represents the cosine of the angle, while the y-coordinate represents the sine.

By understanding the unit circle, you can easily find trigonometric values for various angles. Here's how it works:

• Angles are measured from the positive x-axis, moving counterclockwise.
• Special angles like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\) are commonly used.
• Each of these special angles has well-known sine and cosine values. For example, \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\).

Mastering the unit circle allows you to solve trigonometric functions without relying on a calculator.

Special Angles

Special angles are specific angles commonly used in trigonometry for their simple and known sine and cosine values. These angles include \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \(\frac{\pi}{2}\). They help make calculations easier. Here's why these angles matter:

• They have exact trigonometric values, like \(\sin(\frac{\pi}{6}) = \frac{1}{2}\) and \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\).
• These angles frequently appear in various mathematics problems and applications.
• Knowing these values helps you quickly identify solutions to inverse trigonometric functions.

Whenever you encounter a trigonometric problem, recalling these special angles can greatly simplify your work.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are fundamental in mathematics. They describe relationships between angles and sides of triangles and extend to various real-world applications.

To grasp these functions, consider how they are defined:

• Sine (\(\sin \theta\)): Measures the y-coordinate of the unit circle for a given angle \(\theta\).
• Cosine (\(\cos \theta\)): Measures the x-coordinate of the unit circle for the same angle.
• Tangent (\(\tan \theta\)): Ratio of sine to cosine or \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

These functions reveal patterns and relationships within triangles, enabling us to calculate distances, velocities, and even model waves. When dealing with \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\), knowing the sine values of special angles guides you to quickly find the solution, like \(\frac{\pi}{3}\). Understanding trigonometric functions is crucial for solving these types of problems effectively.