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Chapter 1: Problem 62

Convert each angle to decimal degrees. When necessary round to four decimal places. $$ 45^{\circ} 6^{\prime} $$

Short Answer

Expert verified
45.1 degrees.

Step by step solution

01

Understand the Components of the Angle

The angle given is in degrees and minutes. There are 45 degrees and 6 minutes. Remember that one degree is equal to 60 minutes.
02

Convert Minutes to Degrees

Since 1 degree = 60 minutes, convert the minutes part of the angle to degrees by dividing by 60. So, $$ 6^{\rprime} = \frac{6}{60} \text{ degrees} = 0.1 \text{ degrees} $$.
03

Add Degrees and Decimal Degrees

Now, add the converted decimal degrees to the whole number degrees:$$ 45^{\rprime} + 0.1^{\rprime} = 45.1^{\rprime} $$.

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Key Concepts

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

Degrees and Minutes
When dealing with angles, you often encounter measurements in degrees and minutes. A full circle has 360 degrees. Each degree is further divided into 60 equal parts called minutes. Think of it like time: just as an hour has 60 minutes, each degree has its own 60 minutes. So when you see an angle like 45 degrees and 6 minutes, it signifies 45 full degrees plus 6 additional subdivisions of one degree.
Decimal Degrees
To simplify angles for computation, we often convert them to decimal degrees. This requires translating the minutes into a fraction of a degree. For instance, 6 minutes is a fraction of a degree. To convert minutes to decimal degrees, divide the number of minutes by 60. For example, \( \frac{6}{60} = 0.1 \) degrees. By converting, you simplify mathematical operations, especially in trigonometry and navigation.
Trigonometry Conversion
Trigonometry involves measuring angles and sides of triangles, often using decimal degrees for precision. Techniques like sine, cosine, and tangent rely on accurate angle measurements. Converting degrees and minutes into a decimal form can make trigonometric calculations more straightforward. This ensures better accuracy in fields like engineering, astronomy, and even everyday navigation.

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Most popular questions from this chapter

Solve each problem. Phoenix Temperature The temperature in Phoenix for a day in July is modeled by the function $$ T=18 \sin \left(\frac{\pi}{12}(h-12)\right)+102 $$ where \(h\) is time in hours and \(T\) is degrees Fahrenheit. Find the temperature at \(h=18\) (the daytime high) and at \(h=6\) (the nighttime low).

Solve each problem. Throwing a Javelin The formula $$ d=\frac{1}{32} v_{0}^{2} \sin (2 \theta) $$ gives the distance \(d\) in feet that a projectile will travel when its launch angle is \(\theta\) and its initial velocity is \(v_{0}\) feet per second. Approximately what initial velocity in miles per hour does it take to throw a javelin \(367 \mathrm{ft}\) assuming that \(\theta\) is \(43^{\circ} ?\) Round to the nearest tenth.

Ordered Triples How many ordered triples of real numbers satisfy the following conditions? The product of \(x\) and \(y\) is \(z\). The product of \(x\) and \(z\) is \(y\). The product of \(y\) and \(z\) is \(x\).

Venn Watering Diagram An irrigation specialist draws three circles on the ground of radius 10 meters so that the center of the second circle is on the first circle and the center of the third circle is an intersection point of the first two circles. At the center of each circle he places a sprinkler that waters in a circular pattern with a 10 -meter radius. Find the exact and approximate total area watered by the three sprinklers.

Use a calculator to evaluate each expression. Round approximate answers to four decimal places. $$ 2 \cos \left(-9.5^{\circ}\right)-1 $$
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