Problem 16 - Algebra 2

Question

Find each angle measure to the nearest tenth of a degree. \(\sin ^{-1} 0.052\)

Step-by-Step Solution

Answer

The angle measure to the nearest tenth of a degree is approximately equal to 3.0 degrees.

Step 1: Conversion to Degrees

To begin, apply the inverse sine operation on the provided decimal number. This can be carried out by using a scientific calculator which would give the result in radians, hence convert it into degrees. Remember that the inverse sine function, denoted as \( \sin^{-1} \) or \(\arcsin \), does the opposite of the sine function. It tells you what angle produces the given sine value. This value is usually between -90 and 90 degrees or -π/2 and π/2 if in radians.

Step 2: Perform the Calculation

Enter \( \sin^{-1} 0.052 \) into your scientific calculator. In most cases, your calculator will provide this answer in radians. Hence convert this value into degrees by using the conversion ratio \( \frac{180}{\pi} \)

Step 3: Final Conversion

Obtain the final value to the nearest tenth of a degree as required by the exercise. In most scientific calculators, round the result to the closest tenth.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions can be a bit confusing, so let's break it down. These functions are the opposite of the usual trigonometric functions, such as sine, cosine, and tangent. For example, the inverse sine function, written as \( \sin^{-1} \) or \( \arcsin \), helps us find an angle when we already know the sine value of that angle.
When you see \( \sin^{-1} 0.052 \), it means, "What angle has a sine of 0.052?" This is where inverse functions shine. They are crucial in many fields, like engineering and physics, as they help analyze angles based on ratios. In mathematics, these angles are typically presented in radians, but degrees are often more intuitive for us.

Degree Conversion

Converting radians to degrees is an essential skill if you're dealing with angles. Even though many mathematical calculations use radians, degrees are often easier to understand.
When your calculator gives you an angle in radians, you'll need to convert it if you prefer degrees. The conversion ratio is especially handy:

• Degrees = Radians × \( \frac{180}{\pi} \)

This formula converts the radian measure by multiplying it by \( \frac{180}{\pi} \). This way, you can easily switch from the radian world to the degree world. Remembering this conversion helps you understand and communicate angles in everyday terms.

Scientific Calculator Usage

A scientific calculator is a powerful tool in solving trigonometric problems, like finding an angle using its sine. Here’s how you can use it for angle measurements:
First, you need to enter the sine value into the calculator. Look for the \( \sin^{-1} \) or \( \arcsin \) button. This button will help you find the inverse sine.

• Type in the numerical value, such as 0.052.
• Press the \( \sin^{-1} \) button.

Your calculator might give the answer in radians, depending on its settings. To get the result in degrees, use the radian-to-degree conversion.
Some calculators provide settings to switch the output directly. Be sure to check if that's possible to make it easier for repeated use.