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Chapter 4: Problem 31

Change the given angles to equal angles expressed in decimal form to the nearest \(0.01^{\circ}\). $$15^{\circ} 12^{\prime}$$

Short Answer

Expert verified
The angle in decimal form is \(15.20^{\circ}\).

Step by step solution

01

Understand the Angle Format

The angle is given in degrees and minutes, written as \(15^{\circ} 12^{\prime}\), where \(15^{\circ}\) is the degree part and \(12^{\prime}\) is the minute part. Our goal is to convert this angle into decimal degrees.
02

Recall the Conversion Rate

Recall that 1 degree \((1^{\circ})\) is equal to 60 minutes \((60^{\prime})\). Therefore, to convert minutes into degrees, we divide the number of minutes by 60.
03

Convert Minutes to Decimal Degrees

Convert the \(12^{\prime}\) (minutes) to decimal degrees by using the formula: \(12^{\prime} = \frac{12}{60} = 0.2^{\circ}\).
04

Combine Degrees and Decimal Degrees

Add the converted decimal degrees from the minutes to the degree part: \(15^{\circ} + 0.2^{\circ} = 15.2^{\circ}\).
05

Round to Nearest Hundredth

Since the angle is already 15.2 and there are no further decimal places to consider, rounding it to the nearest hundredth degree gives \(15.20^{\circ}\). Even though it looks similar, for accuracy, it's written as \(15.20\) instead of just \(15.2\).

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Key Concepts

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

Degrees and Minutes
In trigonometry and geometry, angles are not always expressed solely in decimal degrees. Sometimes, you'll come across angles that are represented with degrees and minutes. Minutes are smaller units than degrees, where 1 degree (\(1^{\circ}\)) is equal to 60 minutes (\(60^{\prime}\)). This is particularly useful in tasks like navigation and precise calculations where you need smaller increments than whole degrees.

An angle written as \(15^{\circ} 12^{\prime}\) signifies 15 degrees and 12 minutes. Understanding how to interpret this is crucial for converting it into decimal degrees, which is often necessary for more straightforward calculations and applications.
Angle Conversion Steps
Converting an angle from degrees and minutes to decimal degrees can be done with a straightforward process:
  • Recognize the Parts: The angle \(15^{\circ} 12^{\prime}\) is split into two parts: degrees (\(15^{\circ}\)) and minutes (\(12^{\prime}\)).
  • Conversion Formula: To convert the minutes to decimal form, use the formula: number of minutes divided by 60. This is because 1 degree equals 60 minutes.
  • Calculation: Applying the formula, you get \(\frac{12}{60} = 0.2^{\circ}\). This conversion shows that \(12^{\prime}\) is equal to \(0.2^{\circ}\).
  • Combining Values: Add the result to the degrees to complete the conversion: \(15^{\circ} + 0.2^{\circ} = 15.2^{\circ}\).
Following these steps ensures you convert angles accurately into a decimal format suitable for math operations.
Rounding Decimal Angles
When you've converted an angle into a decimal form, it's often necessary to round it to the nearest hundredth. This ensures precision while keeping the expression simple.

In the example of \(15.2^{\circ}\), you might notice it already appears rounded to one decimal place. To round to the nearest hundredth, you add a zero, making it \(15.20^{\circ}\). This indicates precision even when the numbers beyond the tenths place are zero.
  • Why Round? Rounding is used to simplify numbers while maintaining a degree of accuracy that's fit for practical use.
  • How to Round: Inspect the number at the hundredth place. If there's nothing more to consider, add a zero to express it as \(15.20^{\circ}\) for a precise representation.
Rounding correctly ensures your angles are clear and easy to read, reflecting the necessary precision for various applications.

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Most popular questions from this chapter

Solve the given problems. According to Snell's law, if a ray of light passes from air into water with an angle of incidence of \(45.0^{\circ},\) then the angle of refraction \(\theta_{r}\) is given by the equation \(\sin 45.0^{\circ}=1.33 \sin \theta_{r}\) .Find \(\theta_{r}\)

Solve the given problems. One World Trade Center refers to the main building of the rebuilt World Trade Center in New York City, which was opened in 2014 . A person standing on level ground \(350.0 \mathrm{ft}\) from the base of the building must look upward (from the ground) at an angle of \(78.85^{\circ}\) to see the tip of the spire on top of the building. Use a right triangle to find the height of the building to 4 significant digits. (Does your answer remind you of a famous date in U.S. history?) (This problem is included in memory of those who suffered and died as a result of the terrorist attack of September \(11,2001 .)\)

Solve the given problems. The screen on a certain Samsung Galaxy tablet has 2048 pixels along its length and 1536 pixels along its width. Using a right triangle, find the angle between the longer side and the diagonal of the screen. If the diagonal measures 9.7 in. find the length and width of the screen. (Source: www.samsung.com.)

Solve the given problems. Sketch an appropriate figure, unless the figure is given. A guardrail is to be constructed around the top of a circular observation tower. The diameter of the observation area is \(12.3 \mathrm{m}\). If the railing is constructed with 30 equal straight sections, what should be the length of each section?

Change the given angles to equal angles expressed to the nearest minute. $$47.50^{\circ}$$
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