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Chapter 6: Problem 21

Find the complement and the supplement. $$12^{\circ} 3^{\prime} 14^{\prime \prime}$$

Short Answer

Expert verified
Complement: 鈮�77.94611 degrees, Supplement: 鈮�167.94611 degrees

Step by step solution

01

- Convert to Decimal Degrees

First, convert the given angle from degrees, minutes, and seconds to decimal degrees. The angle is given as \( 12^{\circ} 3^{\prime} 14^{\prime\prime} \). Use the conversion formulas: \( 1^{\prime} = 1/60^{\circ} \) and \( 1^{\prime\prime} = 1/3600^{\circ} \).
02

- Calculate Decimal Degrees

Calculate the decimal degrees as follows: \[ 3^{\prime} = \ \frac{3}{60} = 0.05^{\circ} \] \[ 14^{\prime \prime} = \ \frac{14}{3600} 鈮� 0.00389^{\circ} \] \[ \text{Total} = 12^{\circ} \ + \ 0.05^{\circ} \ +\ 0.00389^{\circ} = 12.05389^{\circ} \]
03

- Find the Complement

The complement of an angle is found by subtracting the angle from \( 90^{\circ} \). Thus, the complement is: \[ 90^{\circ} - 12.05389^{\circ} 鈮� 77.94611^{\circ} \]
04

- Find the Supplement

The supplement of an angle is found by subtracting the angle from \( 180^{\circ} \). Thus, the supplement is: \[ 180^{\circ} - 12.05389^{\circ} 鈮� 167.94611^{\circ} \]

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Key Concepts

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

Decimal Degrees
Decimal degrees are a way of representing angles using a decimal number instead of degrees, minutes, and seconds. It simplifies calculations, especially when adding or subtracting angles.

For example, if you have an angle given as \(12^{\text{掳}} 3^{\text{'}} 14^{\text{''}}\), you need to convert this to decimal degrees. Here鈥檚 how you do it:
  • Divide the minutes by 60 because there are 60 minutes in a degree: \(3^{\text{'}} = \frac{3}{60}= 0.05^{\text{掳}}\).
  • Divide the seconds by 3600 because there are 3600 seconds in a degree: \(14^{\text{''}} = \frac{14}{3600} 鈮� 0.00389^{\text{掳}}\).
  • Add these values to the degrees to get the total decimal degrees: \(12^{\text{掳}} + 0.05^{\text{掳}} + 0.00389^{\text{掳}} 鈮� 12.05389^{\text{掳}}\).
Decimal degrees are helpful for various calculations in geometry, trigonometry, and navigation. Once the angle is in decimal degrees, it's easier to find complements and supplements.
Complementary Angles
Complementary angles are two angles that add up to \(90^{\text{掳}}\). If you know one angle, you can find its complement by subtracting the angle from \(90^{\text{掳}}\).

For instance, if you have the angle \(12.05389^{\text{掳}}\), the complement is:
  • Subtract the angle from \(90^{\text{掳}}\):
  • \(90^{\text{掳}} - 12.05389^{\text{掳}} 鈮� 77.94611^{\text{掳}}\).
Complementary angles are often used in right triangle problems, where one of the angles is \(90^{\text{掳}}\), and you need to find the other two. They are also useful in various geometry and trigonometry problems, simplifying certain equations and calculations. Remember that the sum of complementary angles will always be \(90^{\text{掳}}\).
Supplementary Angles
Supplementary angles are two angles that add up to \(180^{\text{掳}}\). Similar to complementary angles, you can find the supplement of an angle by subtracting it from \(180^{\text{掳}}\).

For example, with an angle of \(12.05389^{\text{掳}}\), the supplementary angle is:
  • Subtract the angle from \(180^{\text{掳}}\):
  • \(180^{\text{掳}} - 12.05389^{\text{掳}} 鈮� 167.94611^{\text{掳}}\).
Supplementary angles are important in various geometric proofs and are frequently found in polygon angle calculations. Knowing that two angles are supplementary can help determine unknown angles in complex geometric shapes. The sum of supplementary angles will always be \(180^{\text{掳}}\), making them fundamental in solving many types of angle problems.

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Most popular questions from this chapter

Use a calculator in Exercises \(81-84,\) but do not use the trigonometric function keys. Given that \(\sin 65^{\circ}=0.9063\) \(\cos 65^{\circ}=0.4226\) \(\tan 65^{\circ}=2.1445\) find the trigonometric function values for \(115^{\circ} .\)

Find the signs of the six trigonometric function values for the given angles. $$-57^{\circ}$$

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. $$y=2 \tan \left(\frac{1}{2} x\right)$$

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$f(x)=\frac{x+4}{x}[4.5]$$

Find the function value. Round to four decimal places. $$\tan 1086.2^{\circ}$$
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