Problem 35 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arcsin \frac{7}{8} $$

Step-by-Step Solution

Answer

The angle in degrees that corresponds to \(\arcsin\frac{7}{8}\), rounded to two decimal places.

Step 1: Understanding the arcsine

The arcsine, denoted as \(\arcsin\) or \(\sin^{-1}\), is the inverse function of the sine. It takes a number between -1 and 1 and outputs an angle which has that number as its sine. Here, we need to find out the angle whose sine value results in the fraction 7/8.

Step 2: Use a calculator to find the arcsine of the fraction

Ensure your calculator is set to degrees mode. Input the fraction 7/8, then use the \(\arcsin\) or \(\sin^{-1}\) function of your calculator to find the output angle in degrees.

Step 3: Round the Result

The result should be rounded off to two decimal places as per the requirement of the exercise. Use the round function on your calculator or manually round your output to the nearest hundredth of a degree.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in understanding angles and their relationships with sides in a triangle. These functions, including arcsine \(\arcsin\), allow us to determine angles when we know the sine, cosine, or tangent value. For instance, if you know the sine of an angle is \(\frac{7}{8}\), you can use the arcsine function to find the angle itself. In this way, inverse trigonometric functions bridge the gap between ratios and angles, providing crucial insights in geometry and trigonometry.

Angles

Angles are fundamental in trigonometry, providing a way to measure rotation or opening between two lines. They are usually measured in degrees or radians. In our context, we are working in degrees, which is the most common unit in high school math. An angle can range from 0 to 360 degrees, depending on the amount of rotation. In this example, after calculating \(\arcsin \frac{7}{8}\), the resulting angle will fall within the standard range for arcsine, which is \(-90\) to \(90\) degrees. Understanding angles is crucial for interpreting results in trigonometric problems.

Sine Function

The sine function is a fundamental part of trigonometry, describing the relationship between an angle in a right triangle and its opposite side divided by its hypotenuse. The sine function takes an angle as input and outputs a value between \(-1\) and \(1\). When working backwards with the arcsine, you're essentially asking: what angle gives a sine value of \(\frac{7}{8}\)? This function is key in many real-life applications, including physics and engineering, where it helps model waves and cycles. Remember, the sine function forms one of the building blocks of trigonometric concepts.

Calculator Usage

Using a calculator correctly can make evaluating trigonometric expressions straightforward. First, set your calculator to the correct mode—degrees or radians—as required by your problem. For our purposes, ensure it's in degrees mode. Input the fraction \(\frac{7}{8}\), then press the \(\arcsin\) or \(\sin^{-1}\) button. The display will show the angle whose sine is \(\frac{7}{8}\). Finally, round this value to two decimal places using your calculator’s rounding function. Calculator proficiency is valuable for quick and accurate trigonometric calculations.