Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\). $$ \tan \theta=4.91 $$

Step-by-Step Solution

Answer

(a) \(78.69^{\circ}\); (b) \(78^{\circ} 41^{\prime}\).

Step 1: Understand the Problem

We need to find the angle \( \theta \) where \( \tan \theta = 4.91 \). This angle must be acute, meaning it is between \( 0^{\circ} \) and \( 90^{\circ} \).

Step 2: Use Inverse Tangent Function

To find \( \theta \), use the inverse tangent (arctan) function: \[ \theta = \tan^{-1}(4.91) \]. This calculation will give us the angle in degrees.

Step 3: Calculate the Angle in Degrees

Using a calculator, compute \( \theta = \tan^{-1}(4.91) \). This should give you \( \theta \approx 78.69006753^{\circ} \).

Step 4: Round to Nearest Hundredth of a Degree

For part (a), round \( 78.69006753^{\circ} \) to the nearest hundredth of a degree. Thus, \( \theta \approx 78.69^{\circ} \).

Step 5: Convert Degrees to Minutes

To complete part (b), convert the decimal part of the degrees into minutes. Since there are 60 minutes in a degree, calculate:\( 0.69006753 \times 60 \approx 41.404 \).

Step 6: Round to the Nearest Minute

Round the minutes value from Step 5 to the nearest whole number. This gives us approximately \( 41^{\prime} \).

Step 7: Verify the Calculations

Double-check the calculations for accuracy. Ensure that all the rounding has been done correctly according to the respective parts of the question.

Key Concepts

Inverse Tangent Function

The inverse tangent function, often denoted as arctan or \( \tan^{-1} \), is crucial when you need to find an angle from a given tangent value. In trigonometry, the tangent of an angle is the ratio of the opposite side to the adjacent side in a right triangle. However, sometimes, you know the tangent value and need to find the actual angle. This is where the inverse tangent function comes into play.

The inverse tangent function essentially "reverses" what the tangent does. So when you have \( \tan \theta = 4.91 \), using the inverse tangent function, you retrieve the corresponding angle \( \theta \). In this problem, applying \( \tan^{-1}(4.91) \) gives you the measure of angle \( \theta \).

This function is particularly useful in problems involving acute angles. The range of the inverse tangent function in degrees is \( 0^{\circ} \) to \( 90^{\circ} \), which matches the range of acute angles. Hence, it seamlessly solves angle determination problems where tan values are provided.

Radian and Degree Conversion

Degrees and radians are two units used to measure angles. Often in trigonometry and geometry, you may need to convert between these systems to solve problems appropriately.

Degrees are what's commonly used, where a full circle is 360 degrees. Radians are used more frequently in higher mathematics involving calculus. In terms of radians, a full circle has \( 2\pi \) radians. This gives the conversion formula:

• To convert degrees to radians: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)
• To convert radians to degrees: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \)

In this specific problem, the angle \( \theta = 78.69^{\circ} \), already found in degrees, requires no conversion if the goal is to represent in degrees or minutes. However, if radians were needed or provided, those conversions would be vital.

Angle Approximation

When working with angles, there are often requirements to present angles in a specified form or precision, such as to the nearest hundredth of a degree or in minutes. These approximations are essential for practical applications and precise calculations.

In this exercise, once the angle \( \theta \) is calculated using the inverse tangent function, it is initially in a decimal form of degrees: \( \theta \approx 78.69006753^{\circ} \). For precise applications, it's typically approximated:

• **(a)** rounded to the nearest hundredth of a degree: \( 78.69^{\circ} \)
• **(b)** further converted and rounded to minutes. Since 1 degree is 60 minutes, the decimal part (0.69) must be multiplied by 60. This gives approximately 41 minutes.

Rounding requires understanding both the concept of significant figures and the practicality of converting between units. These approximations are standard practice, ensuring results are both accurate and usable in various mathematical and practical contexts.