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Chapter 1: Problem 9

Convert each angle measure to decimal degrees. $$ 183^{\circ} 33^{\prime} 36^{\prime \prime} $$

Short Answer

Expert verified
The decimal degree equivalent of \(183^{\circ} 33^{\prime} 36^{\prime \prime}\) is 183.56掳.

Step by step solution

01

Convert the minutes to degrees

In this step, convert the given minutes to degrees. Remember, 1 minute is equal to \( \frac{1}{60} \) of a degree. So, multiply the number of minutes with \( \frac{1}{60} \). \n 33' * \( \frac{1}{60} \) = 0.55 degrees.
02

Convert the seconds to degrees

Similar to the first step, convert the seconds to degrees. Bear in mind, 1 second is equal to \( \frac{1}{3600} \) of a degree. Accordingly, multiply the number of seconds with \( \frac{1}{3600} \). \n 36'' * \( \frac{1}{3600} \) = 0.01 degrees.
03

Total up the degrees

Now, sum up the initial degrees, the degrees converted from minutes, and the degrees converted from seconds to obtain the final decimal degrees. \n 183掳 + 0.55掳 + 0.01掳 = 183.56掳.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle Conversion
In geometry and trigonometry, angles are often denoted in degrees, but they can be represented in different formats such as degrees, minutes, and seconds or as decimal degrees. Understanding angle conversion is essential for precise calculations. An angle given in degrees, minutes, and seconds (DMS) can be converted to decimal degrees to simplify computations. To convert, follow these steps:
  • First, convert the minutes (') of the angle to degrees by dividing by 60, as there are 60 minutes in one degree.
  • Next, convert the seconds ('') by dividing by 3600, because there are 3600 seconds in a degree (60 seconds per minute, and 60 minutes per degree).
  • Finally, sum the converted values to the original degree measurement.
This way, an angle like \( 183^{\circ} 33^{\prime} 36^{\prime\prime} \) will become \( 183.56^{\circ} \) when expressed in decimal degrees.
Degree Calculation
Degree calculation is a fundamental skill when dealing with angles. It's crucial to perform arithmetic operations efficiently and accurately, especially when converting angles into different units or conducting trigonometric evaluations. Originally, angles can be given in whole degrees or in a more precise composite form of degrees, minutes, and seconds (dms). To perform a degree calculation:
  • Convert minutes and seconds into degree fractions. Use the value 1/60 for minutes and 1/3600 for seconds to convert these units properly into a degree format.
  • Ensure each step adds correctly to the sum. Include the initial degree measure, the converted minutes, and seconds, resulting in a single decimal degree number.
Such calculations make trigonometric analysis more manageable, providing a simple, singular decimal degree value to work with.
Trigonometric Concepts
Understanding trigonometric concepts involves grasping how angles, most often expressed in degrees, can interact with functions like sine, cosine, and tangent. Whether operating in degrees or radians, converting angles into decimal form is beneficial for these calculations. For trigonometric operations:
  • Ensure conversions between DMS and decimal degrees are correctly applied. This allows for seamless application in trigonometric functions.
  • Accurate degree representation is critical, as functions like sine and cosine are sensitive to angle measurement precision.
Once you have an angle in decimal degrees form, use trigonometric tables or calculators to find values of sine, cosine, or tangent. Mastering these conversions and calculations substantially aids in solving various mathematical and real-world problems.

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Most popular questions from this chapter

A small airplane flying at an altitude of 5300 feet sights two cars in front of the plane traveling on a road directly beneath it. The angle of depression to the nearest car is \(62^{\circ}\) and the angle of depression to the more distant car is \(41^{\circ} .\) How far apart are the cars?

Convert each angle measure from radians to degrees. $$ \frac{\pi}{3} $$

Use the information given to find the other two trigonometric ratios. $$ \tan \theta=1.5 $$

Use the information given to find the other two trigonometric ratios. $$ \cos \theta=\frac{1}{2} $$

Round answer to the nearest \(10^{t h}\). A man standing on the roof of a building 70 feet high looks at the building next door. The angle of depression to the roof of the building next door is \(36^{\circ}\). The angle of depression to the bottom of the building next door is \(65^{\circ} .\) How tall is the building next door?
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