Problem 33 - Algebra and Trigonometry

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$ \sin ^{-1}\left(\sin \frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

\(\sin ^{-1}(\sin (\pi/3)) = \pi/3\)

Step 1: Evaluate the sine function

The sine of \(\pi/3\) is already a known standard angle in the unit circle, it's equal to \(\sqrt{3}/2\). Thus, the expression can be rewritten as \(\sin ^{-1}(\sqrt{3}/2)\).

Step 2: Find the arcsin

The arcsin function is the inverse operation of the sin function. Here, we need to find the angle in the interval from \(-\pi/2\) to \(\pi/2\), whose sine is \(\sqrt{3}/2\). Which is precisely \(\pi/3\).

Key Concepts

Unit Circle

The unit circle is a critical tool in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This simple setup is useful because each point on the circle corresponds to an angle and can describe the sine and cosine of that angle.

• The circle's radius intersects the coordinate axes at four key points: (1,0), (-1,0), (0,1), (0,-1).
• These intersections are related to common angles: 0, \(rac{\pi}{2}\), \(rac{3\pi}{2}\), and \(2\pi\).

These points and angles help us to evaluate trigonometric functions such as sine and cosine without calculators.

Angles are measured in radians, which is a system based on the arc length of the circle. It results in values like π/3, which are easier to work with for exact trigonometric evaluations. Understanding this framework is key in evaluating trigonometric functions and their inverses.

Sine Function

The sine function is one of the fundamental trigonometric functions that describes the relationship between an angle and the vertical component of a point on the unit circle.

• The sine of an angle, say \(\theta\), is defined as the y-coordinate of the point where the terminal side of the angle intersects the unit circle.
• For standard angles like \(\pi/3\), the sine value is \(\sqrt{3}/2\). This is derived from the special triangles, specifically the 30-60-90 triangle.

Sine values are periodic, meaning they repeat after a complete cycle of \(2\pi\) radians. This periodic nature is crucial as it defines the behavior of the sine function over its domain.

The sine function is strictly increasing from \(-\pi/2\) to \(\pi/2\), which is known as its principal range. Within this range, it covers all possible output values from -1 to 1, making it possible for us to define an inverse function.

Arcsine

Arcsine, denoted as \(\sin^{-1}\), is the inverse of the sine function. It takes a sine value and returns the corresponding angle whose sine is that value.

• The output of arcsine is an angle within the interval \([-\pi/2, \pi/2]\).
• Each sine value yields a single corresponding angle in this interval.

For example, if you know that the sine of \(\pi/3\) is \(\sqrt{3}/2\), then \(\sin^{-1}(\sqrt{3}/2)\) returns \(\pi/3\). This is because \(\pi/3\) is the angle in the principal range whose sine is \(\sqrt{3}/2\).

Understanding arcsine is essential for solving equations where you need to find an angle given its sine. It is important to remember that arcsine only works with values between -1 and 1, reflecting the range of the sine function.

Exact Values

In trigonometry, especially when dealing with functions like sine and arcsine, working with exact values is crucial for solving expressions without a calculator.

• Exact values are often tied to angles that are commonly found in geometric shapes, such as 30°, 45°, and 60° in degrees, or \(\pi/6\), \(\pi/4\), and \(\pi/3\) in radians.
• These values stem from special triangles, like the 30-60-90 and the 45-45-90 triangles, which have well-defined sine, cosine, and tangent values.

Using exact values is beneficial because they provide precise answers and understanding of trigonometric functions.

For example, when asked for \(\sin^{-1}(\sin(\pi/3))\), knowing that the exact sine value of \(\pi/3\) is \(\sqrt{3}/2\) allows you to correctly determine that the answer is \(\pi/3\) without reliance on approximation or a calculator.