Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\). $$ \sin \theta=0.4217 $$

Step-by-Step Solution

Answer

(a) \( \theta \approx 24.96^{\circ} \); (b) \( \theta \approx 24^{\circ}\, 58' \).

Step 1: Identify the Function Involved

We're informed that \( \sin(\theta) = 0.4217 \). This means we need to find \( \theta \) using the arcsine function or \( \sin^{-1} \).

Step 2: Use the Inverse Sine Function

To find \( \theta \), take the inverse sine of 0.4217:\[ \theta = \sin^{-1}(0.4217) \]Use a calculator to get the approximate value in degrees.

Step 3: Approximate to the Nearest 0.01°

Using a calculator, \( \sin^{-1}(0.4217) \) gives approximately \( 24.9641365372^{\circ} \). Round this to the nearest \( 0.01^{\circ} \):\[ \theta \approx 24.96^{\circ} \]

Step 4: Convert to Minutes for Approximating to the Nearest 1′

Convert the decimal degrees \( 24.9641365372^{\circ} \) to degrees and minutes. The decimal part after subtracting 24 is \( 0.9641365372 \). Convert this into minutes:\[ 0.9641365372 \, \text{degrees} \times 60 = 57.848192232 \, \text{minutes} \]Round to the nearest minute:\[ \theta \approx 24^{\circ} \, 58' \]

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given specific trigonometric values. These functions reverse the action of the regular trigonometric functions, moving from ratio to angle instead of angle to ratio. If you know the sine, cosine, or tangent of an angle, you can find the angle itself using the corresponding inverse function. For example, the inverse sine function, denoted as \( \sin^{-1} \) or arcsin, is used to determine the angle whose sine is a given number. In our exercise, we used \( \sin^{-1} \) to determine that when \( \sin(\theta) = 0.4217 \), the angle \( \theta \) is approximately \( 24.96^{\circ} \). This is one of the main tools for solving problems involving unknown angles.

Angle Measurement

Angles can be measured in several ways, the most common being degrees and radians. A full rotation around a circle is \( 360^{\circ} \) or \( 2\pi \) radians. In everyday situations, degrees are more common and are often subdivided further into minutes and seconds for more exact measurements. In the given exercise, we first rounded the angle to the nearest 0.01 degree. Degrees can be easily split into smaller pieces:

• 1 degree = 60 minutes (\(1^\circ = 60'\)).
• 1 minute = 60 seconds (\(1' = 60"\)).

Subdividing angles helps achieve the precision needed in fields like astronomy and navigation. By converting the decimal part of the angle \( 24.9641365372^{\circ} \) to minutes, we were able to round it to \( 24^{\circ} \, 58' \). This demonstrates how these finer measurements provide a more accurate representation of an angle.

Radian and Degree Conversion

Radian and degree conversion is crucial for solving trigonometric problems because different contexts may require different units. While degrees are more intuitive for everyday usage, radians are often used in calculus and advanced mathematics since they result in more straightforward derivatives and integrals. To convert from degrees to radians, we use the relation:\[\text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right)\]Conversely, converting from radians to degrees involves multiplying by \( \frac{180}{\pi} \). While the exercise primarily involved degrees, understanding this conversion allows the flexibility needed in various math applications. For example, converting \( 24.96^{\circ} \) to radians involves:\[24.96^{\circ} \times \left( \frac{\pi}{180} \right) \approx 0.4357 \, \text{radians}\]This conversion is integral to fluently moving between different mathematical systems and solving problems with ease.