Problem 35 - Precalculus Essentials

Question

$$\text {use a calculator to find the value of the acute}\text { angle } \theta \text { to the nearest degree.}$$ $$\sin \theta=0.2974$$

Step-by-Step Solution

Answer

The acute angle \(\theta\) to the nearest degree is approximately 17°.

Step 1: Understanding the Problem

Given that \(\sin \theta = 0.2974\). We are tasked to find the value of \(\theta\), which is the angle whose sine value is 0.2974.

Step 2: Calculating the Angle Value

Take the arcsine (inverse sine or \(\sin^{-1}\)) of the given sine value. This can be done using a scientific calculator. The order to enter values into the calculator is ``arcsin(0.2974)``.

Step 3: Rounding the Angle

The calculator gives a value that may have many decimal places. Round this value to the nearest whole number to get the acute angle \(\theta\) in degree.

Key Concepts

Inverse Sine Function

The inverse sine function, often denoted as \( \sin^{-1} \) or arcsine, is a key topic in trigonometry. This function is used when we need to find the angle that corresponds to a specific sine value.
For example, when given \( \sin \theta = 0.2974 \) as in our exercise, using the inverse sine gives us the angle \( \theta \). Essentially, if \( \sin A = x \), then \( A = \sin^{-1}(x) \) for angles, resulting in \( A \) as the output, which represents the angle whose sine is \( x \).

Key points to remember about inverse sine are:

• The input must always be a value between -1 and 1, as those are the possible outputs for \( \sin \theta \).
• The output will be an angle, which falls within certain bounds, typically between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians, or for simplicity, between -90 and 90 degrees.
• In terms of right triangles, this function finds the angle when you know the lengths related to opposite side and hypotenuse.

Understanding this fundamental aspect of trigonometric functions is crucial for solving problems that involve determining angles from trigonometric ratios.

Angle Measurement

Angle measurement is conducted in degrees or radians. For many practical applications in trigonometry, degrees are used due to their intuitive nature.
When you calculate the angle \( \theta \) using a scientific calculator after determining \( \sin^{-1}(0.2974) \), the result is typically given in degrees because it's more commonly used outside of advanced mathematics.

To round an angle, consider the following:

• Check the value immediately following the decimal point. If it's 5 or more, round up. Otherwise, round down.
• This precision to the nearest degree ensures easy interpretation and use, particularly in fields like engineering, navigation, and architecture.

Obtaining the angle in degrees allows for practical applications and easier communication of angular measures in daily and academic situations.

Scientific Calculator Usage

A scientific calculator is an essential tool for solving trigonometric equations, such as finding angles using inverse functions.
When using a calculator to find \( \theta \) from \( \sin^{-1}(0.2974) \), it's important to follow these steps:

• Ensure the calculator is in degree mode, as many calculators default to radians.
• Input the function by pressing the "sin^-1" or "arcsin" button.
• Enter the given sine value, here 0.2974, directly into the calculator and close the parenthesis if required.

The calculator will provide a numerical amount that represents the angle in degrees. For example, you might see a result like 17.299 degrees, which you would then round to 17 degrees as a final answer.
Understanding how to navigate your calculator and set the correct mode is crucial for ensuring accurate mathematical computations.