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Chapter 6: Problem 34

Convert to decimal degree notation. Round to two decimal places. $$64^{\circ} 53^{\prime}$$

Short Answer

Expert verified
64.88^{\circ}

Step by step solution

01

Understanding the Components

Identify the components of the angle given in degrees, minutes, and seconds. Here, you have 64 degrees and 53 minutes.
02

Convert Minutes to Degrees

Convert the minutes to degrees. One minute is equal to \(\frac{1}{60}\) of a degree. Therefore, you need to multiply the number of minutes by \(\frac{1}{60}\): \(53^{\prime} = 53 \times \frac{1}{60} = 0.883333\ldots \) degrees.
03

Add the Degrees

Add the degrees from the original angle to the degrees obtained by converting the minutes: \(64^{\circ} + 0.883333\ldots^{\circ} \).
04

Round to Two Decimal Places

Finally, round the result to two decimal places: \(64.883333\ldots \approx 64.88^{\circ} \).

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Key Concepts

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

Angle Conversion
Many students often encounter problems when converting angles between different notations. This process involves understanding minute parts of an angle and converting them into a single notation, typically in decimal degrees.
In mathematical contexts, angles can be expressed in various notations, such as degrees, minutes, and seconds. These notations are often used in surveying, astronomy, and engineering.
For ease of calculation and to ensure consistency, it is ideal to convert mixed notation into a single standard, which is often decimal degrees. This conversion simplifies further mathematical operations and comparisons directly related to angles.
Degrees and Minutes
Degrees and minutes are units used to measure angles. It is vital to understand the relationship between these units to convert them accurately.
One degree (\(1^{\bigcirc}\) ) is divided into 60 minutes (\(1^{'}\) ). Thus:
  • \(1^{\bigcirc} = 60^{'}\)
  • \(1^{'} = \frac{1}{60} \text{ degree}\)
To convert minutes to degrees, you multiply the number of minutes by \(\frac{1}{60}\).
In our exercise, 53 minutes is converted as follows:
\({53^{'} = 53 \times \frac{1}{60} = 0.883333\text{... degrees}}\).
This step decomposes complex mixed notation into a single measurable unit, simplifying the addition of angles.
Rounding Numbers
Rounding is a method used to simplify numbers, making them easier to work with while retaining a close approximation.
In our context, we are converting angles to a decimal degree notation and then rounding to two decimal places. This involves:
  • Identifying the third decimal place value.
  • If the third decimal is 5 or greater, increase the second decimal by one.
  • If it鈥檚 less than 5, simply retain the second decimal without change.
For our converted angle:
64.883333... we look at the third decimal (3), and since it is less than 5, we leave the second decimal (8) unchanged.
Thus, rounded to two decimal places, the angle is \(64.88^{\bigcirc}\).
This simplification helps in making calculations and comparisons more manageable.

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Most popular questions from this chapter

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=\frac{1}{2} \sec \left(\frac{1}{2} x\right)$$

Given that \(\sin 38.7^{\circ} \approx 0.6252, \cos 38.7^{\circ} \approx 0.7804\) and \(\tan 38.7^{\circ} \approx 0.8012,\) find the six function values of \(51.3^{\circ}\).

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=-2+\cot x$$

Find the function value. Round to four decimal places. $$\csc 520^{\circ}$$

Find the signs of the six trigonometric function values for the given angles. $$290^{\circ}$$
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