Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to give each value of \(\theta\) in decimal degrees. $$\theta=\arccos (-0.39876459)$$

Step-by-Step Solution

Answer

The angle \(\theta\) is approximately \(113.496\) degrees.

Step 1: Understand the Problem

Given the trigonometric function \(\theta = \arccos(-0.39876459)\), we need to find the angle \(\theta\) in decimal degrees.

Step 2: Input the Value into a Calculator

To solve \(\theta = \arccos(-0.39876459)\), input the value \(-0.39876459\) into the calculator using the arccosine function (often denoted as \(\cos^{-1}\) on calculators). Ensure that the calculator is set to degree mode to get the answer in decimal degrees.

Step 3: Obtain the Result

Upon calculating, the calculator should display the value of \(\theta\) in decimal degrees. This will be the angle whose cosine is closest to \(-0.39876459\). Based on typical calculator results, this angle is approximately \(113.496\) degrees.

Key Concepts

Arccosine

Arccosine, often denoted as \( \cos^{-1} \) or \( \text{arccos} \), is a trigonometric function used to find an angle when its cosine value is known. In simpler terms, if you know that the cosine of an angle is a particular numeric value, you can use arccosine to work backwards and find the angle itself.

• Arccosine gives the principal value of an angle, which is always between 0 and 180 degrees in terms of its range.
• Mathematically, if \( \theta = \arccos(x) \), then it implies that \( x = \cos(\theta) \).
• It is commonly used in problems where the result must provide geometric conformations or resolve calculations involving angles from scalar dot products.

The challenge with using the arccosine function is ensuring the input value is between -1 and 1, as these are the only valid inputs for cosine values. If you attempt to compute \( \arccos(x) \) where \( x \) is outside this range, the calculator or mathematical software will give an error.

Decimal Degrees

Decimal degrees is a numerical expression of angles, where degrees are presented in a decimal format rather than in the degree-minute-second (DMS) format. This method simplifies certain calculations and makes it easier to read computational results.

• In decimal degrees, the full circle is still 360 degrees, similar to the DMS format.
• Decimal degrees do not use arcminutes or arcseconds and purely represent fractions of a degree.

To convert from DMS to decimal degrees, you can use this simple calculation:Convert all components to degrees (e.g., minutes and seconds need to be divided by 60 and 3600, respectively). Sum these up to get the angle in decimal degrees. For example, an angle of 30 degrees 15 minutes 22 seconds can be converted as:\[ 30 + \frac{15}{60} + \frac{22}{3600} \approx 30.2561 \text{ degrees} \]Decimal degrees make it particularly easy when you utilize them for trigonometric calculations using technology, such as computers or scientific calculators.

Calculator Use

Using a calculator effectively for trigonometric functions like arccosine is crucial in both academic and practical applications. Here’s a short guide on how to make sure you're getting accurate results:

• First, ensure your calculator is set to the correct mode. For angle calculations in degrees, the calculator should be in 'degree mode'.
• Most calculators have a specific button for arccosine, often labeled as \( \cos^{-1} \) or "arc cos". Double-check your calculator's manual to find the correct input method.
• Input the value for which you need to determine the angle, ensuring that it lies between -1 and 1, as these are the only valid input range for the cosine function.
• Finally, verify the result to ensure that it matches expected outcomes; for instance, knowing roughly the domain for the given input can help confirm if the output is reasonable.

If using a scientific calculator app or online software, the process is usually similar, though the interface may differ. Familiarizing yourself with the software or hardware will help you optimize the accuracy and efficiency of the calculations you perform.