Problem 54 - Algebra 2

Question

Find each angle measure to the nearest tenth of a degree. $$ \sin ^{-1} \frac{5}{8} $$

Step-by-Step Solution

Answer

The angle measure is approximately 38.7 degrees.

Step 1: Understand the problem

The problem is asking to find the angle whose sine is 5/8. The sine of an angle (in a right triangle) is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. This means we are looking for an angle in a right triangle where the length of the opposite side is 5 units and the length of the hypotenuse is 8 units.

Step 2: Use the inverse sine function

The inverse sine function, denoted as \( \sin^{-1} \) or arcsin, is used to find the measure of an angle when the sine of the angle is known. Here, we are given that \( \sin \theta = \frac{5}{8} \). To find \( \theta \), we use \( \theta = \sin^{-1} \frac{5}{8} \).

Step 3: Calculate the angle

Use a scientific calculator to calculate the inverse sine of 5/8. Make sure your calculator is set to degree mode since we want the answer in degrees. Key in the values and calculate, then round to the nearest tenth.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, relating angles to ratios of sides in right triangles. Here’s how it works:

• The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.
• It's mathematically expressed as: \( ext{sin}( heta) = \frac{ ext{opposite}}{ ext{hypotenuse}} \).

For example, if you have a triangle with an angle \( \theta \), where the opposite side is 5 units long and the hypotenuse is 8 units long, the sine of that angle is \( \frac{5}{8} \).In the context of inverse trigonometric functions, when you know this ratio, the inverse sine function \(\sin^{-1}\) helps you find the actual angle \(\theta\). This is especially useful in calculations where you need to determine specific angle measures, just like in the problem you've inquired about.

Angle Measurement

Angles are a measure of rotation and are quantified in degrees or radians. For solving trigonometric problems like the one given, it's essential to understand:

• Degrees are the most common unit for angle measurement in everyday applications.
• A full circle is 360 degrees, meaning a right angle is 90 degrees.
• When using a calculator, ensure it is set to the correct unit (degrees) to avoid errors in calculations.

In the original exercise, the task is to find the angle in degrees using the inverse sine function. This ensures that the angle falls within the customary range of values for practical scenarios, simplifying its application in real-world problems.

Right Triangle Trigonometry

Right triangle trigonometry deals with the relationships between the angles and sides of right-angled triangles. Understanding these concepts is crucial for solving trigonometric problems.Key principles include:

• Each right triangle has one 90-degree angle.
• The two other angles must add up to 90 degrees.
• The side opposite the 90-degree angle is the hypotenuse, the longest side.
• The sides forming the right angle are the opposite and adjacent sides relative to a chosen angle \(\theta\).

When working with problems, such as finding \( \theta \) using \( \sin^{-1} \frac{5}{8} \), you apply the trigonometric functions specific to right triangles. By understanding these basic properties, you can navigate more complex problems involving triangles and their angles with ease.