Problem 5 - Precalculus with Limits

Question

Evaluate the expression without using a calculator. \(\arcsin \frac{1}{2}\)

Step-by-Step Solution

Answer

The value of \( \arcsin \frac{1}{2} \) is \( \frac{π}{6} \).

Step 1: Recall the standard angles and their sine values

The sine of an angle is equal to 1/2 at two standard angles: \( \frac{π}{6} \) (or 30 degrees) and \( \frac{5π}{6} \) (or 150 degrees) in one period of sine function. However, the range of arcsine function is from -\( \frac{π}{2} \) to \( \frac{π}{2} \), so the angle which is in this range and has a sine value of 1/2 is frst one.

Step 2: Find the arcsine

Therefore, \( \arcsin \frac{1}{2} = \frac{π}{6} \).

Key Concepts

Standard Angles in Trigonometry

Standard angles in trigonometry refer to specific, commonly used angles that are generally found in a right-angled triangle. The sine, cosine, and tangent values of these angles are often memorized to help solve various trigonometric problems without a calculator.

The standard angles are usually measured in degrees (°) or radians (rad). They typically include 0° or 0 rad, 30° or \( \frac{\pi}{6} \) rad, 45° or \( \frac{\pi}{4} \) rad, 60° or \( \frac{\pi}{3} \) rad, and 90° or \( \frac{\pi}{2} \) rad. Knowing the sine, cosine, and tangent values for these angles simplifies many trigonometric calculations. For instance, the sine of 30° or \( \frac{\pi}{6} \) is 1/2, and this knowledge is crucial in solving the given exercise.

Sine Values of Standard Angles

The sine function relates a given angle to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. For standard angles, sine values are often remembered as part of the trigonometric landscape.

Here's a simple way to remember the sine values for key standard angles:

• Sine of 0° or 0 rad: 0
• Sine of 30° or \( \frac{\pi}{6} \) rad: \( \frac{1}{2} \)
• Sine of 45° or \( \frac{\pi}{4} \) rad: \( \frac{\sqrt{2}}{2} \)
• Sine of 60° or \( \frac{\pi}{3} \) rad: \( \frac{\sqrt{3}}{2} \)
• Sine of 90° or \( \frac{\pi}{2} \) rad: 1

Understanding these key sine values is essential when evaluating trigonometric expressions like \( \arcsin \frac{1}{2} \), where recognizing that the sine of \( \frac{\pi}{6} \) rad is 1/2 leads to the solution.

Range of the Arcsine Function

The arcsine function, also denoted as \( \arcsin \) or \( \sin^{-1} \), is the inverse of the sine function. Since the sine function only has values between -1 and 1, the arcsine function is defined only for this interval. Moreover, to be a proper function—giving one output for each input—the arcsine function has a restricted range.

The range of the arcsine function is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), also equal to -90° to 90°. This means when you evaluate \( \arcsin \frac{1}{2} \), the angle you're looking for must fall within that range. Considering the standard angles, \( \frac{\pi}{6} \) is the angle in the restricted domain where the sine equals 1/2, leading to the result \( \arcsin \frac{1}{2} = \frac{\pi}{6} \). It's important to remember this range when solving problems involving the arcsine, as it helps determine the correct angle.