anglecalc

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Anglecalc.doc June 7, 2006

ANGLE CALCULATIONS

An angle is formed by rotating a given ray about its endpoint to some terminal position. The original ray is the initial side of the angle and the second ray is the terminal side of the angle. The common endpoint is the vertex of the angle (see figure below). The measure of an angle is determined by the amount of rotation of the terminal ray from the initial ray. For the purposes of this worksheet we will discuss two ways to measure angles: by degrees and by radians. Degrees The concept of measuring angles in degrees grew out of the belief of the early Babylonians that the seasons repeated every 360 days. One degree is the measure of an angle formed by rotating a ray (one three hundred sixtieth) of a complete revolution. There are two popular methods for representing degrees and their fractional parts. One is the decimal degree method. For example, the measure 29.76° is a decimal degree. It means

°29 plus 76 hundredths of 1°

A second method of measurement is known as the DMS (Degree, Minute, Second) method. For example, the measure 126° 12′ 27″ is a degree value expressed in DMS form. It means

126° plus 12 minutes plus 27 seconds In the DMS method the fractional part of a degree may be expressed by understanding that we subdivide a degree into 60 equal parts, each of which is called a minute (denoted by ′) and that a minute is subdivided into 60 equal parts, each of which is called a second (denoted by ″). Thus 1º = 60′, 1′ = 60″, and 1º = 3600″. Changing Minutes and Seconds to Decimal Degrees: It is sometimes necessary to change minutes or seconds to decimal equivalents or vice versa. Minutes or seconds are first changed to their fractional part of a degree. Then the fraction is changed to its decimal equivalent by dividing the numerator by the denominator.

360011 and degree,a of

6011 :Remember =′′=′

vertex

initial side

Figure 1

Angle Calculations 2


• To change minutes to a decimal part of a degree: Divide minutes by 60. • To change seconds to a decimal part of a degree: Divide seconds by 3600.

For example: Convert 50° 15′ 27″ to a decimal degree value. Change 15′ to its decimal degree equivalent . = 0.25º

Change 27″ to its decimal degree equivalent. = 0.0075°

And then add the values together: 50°+ 0.25°+ 0.0075º = 50.2575° Changing a Decimal Degree into a DMS Degree Value: The decimal part of a degree can be changed to minutes and seconds by reversing the procedure. To change a decimal part of a degree to minutes, multiply by 60. Similarly, to change the decimal part of a minute to seconds, multiply by 60.

• To change a decimal part of a degree to minutes: Multiply the decimal part of a degree by 60. • To change a decimal part of a minute to seconds: Multiply the decimal part of a minute by 60.

For example: Convert 50.75° into a DMS degree value Change 0.75° to minutes 0.75 × 60 = 45′ And so... 50.75° = 50° 45′ For example: Convert 28.43° into a DMS degree value Change 0.43° to minutes and seconds 0.43 × 60 = 25.8' (degrees to minutes) 0.8 × 60 = 48' (decimal part of min to sec) And so... 28.43° = 28° 25' 48″ Adding and Subtracting Angle Measures: Angle measures can be added or subtracted. Keep in mind that only like measures can be added or subtracted. To add, arrange the measures in columns of like measures. For example: which simplifies* to 94° 44' 13″

*Since 73″ = 1' 13″, we must simplify and shift the values to the left as appropriate. In this case

we must add one minute to 43 to make 44 minutes leaving a remainder of 13 seconds.

156027

3600

12 15 5482 28 1994 43 73

°+ °

°

' "' "' "

Angle Calculations Worksheet


3

To subtract, arrange the measures in columns of like measures; borrow as needed. (Borrow 1° from 37°. 1° = 60′ and 60′ + 15′ = 75′.)

For example: Since we can’t subtract 32 from 15 with real numbers, we must borrow (much like you would in regular whole number subtraction) one degree to make more minutes from which to subtract. Complementary & Supplementary Angles: If the sum of the measures of two angles equals one straight line (180°), the angles are called supplementary. If the sum of the measures of two angles equals one right angle (90°), the angles are called complementary. To find the complement of any angle, subtract the angle from 90°; to find the supplement of any angle, subtract the angle from 180°. For example: Find the complement of 63° 37'. (Borrow 1° from 90°. 1° = 60′ )

Multiplying and Dividing Angle Measures: To multiply or divide angle measures, perform the indicated operation and simplify as needed. For example: An angle whose measure is 65° 02' 37" needs to be twice as large. Find the measure of the new angle. (65° 02' 37") * 2 = 130° 04' 74". Since 74" = 1' 14", we must simplify to a final answer of 130° 05' 14".

PRACTICE: Perform the indicated operations. Be sure to simplify your final answers. 1. Change 0.42° to equivalent minutes and seconds.

90 89 6063 37 63 37

26 23

° = °− ° = − °

°

'' '

'

34 21

23 15- 23 15-57 36 51 37

′°

′°=′°

′°=′°



4

2. Change 15° 4' to its decimal degree equivalent rounded to the nearest ten-thousandth. 3. Change 0.46° to equivalent minutes and seconds. 4. Change 8° 20' to its decimal degree equivalent rounded to the nearest ten-thousandth. 5) Add and simplify: 6) Subtract and simplify: 7) Add and simplify: 8) Subtract and simplify: 9. Find the measure of an angle with a complement of 35°.

45" 12' 37 18" 47' 15

°+°

23" 35' 114 28' 147

°−°

55" 8' 7 14" 10' 45

°+°

8" 10' 20 6" 32

°−°



5

10. Find the measure of an angle with a supplement of 35°. 11. An angle whose measure is 17° 36' 40" needs to be three times as large. Find the measure of the new angle in degrees and minutes. 12. An angle whose measure is 45° 37' 30" needs to be twice as large. Find the measure of. the new angle in degrees and minutes. 13. A right angle will be divided into four equal angles. Find the measure of each new angle in degrees and minutes. 14. Find the complement of 40° 37' 26". Then convert the result to its decimal equivalent

rounded to the nearest ten-thousandth. ANSWER KEY 1. 25' 12" 2. 15.0667° 3. 27' 36" 4. 8.3333° 5. 53° 3" 6. 32° 52' 37" 7. 52° 19' 9" 8. 11° 49' 58" 9. 55° 10. 145° 11. 52° 50' 12. 91° 15' 13. 22° 30' 14. 49.3761°



6

Radians Until now, we have been measuring angles in terms of degrees, minutes and seconds, or in degrees and decimal parts of degrees. In order to deal with sinusoidal waveforms and some other important ideas of physics and calculus, another system of angle measurement is necessary, namely, radians. Technically, at one radian the measure of the radius (half the distance across any particular circle) is equal to the arc length marked by the intersection of the two rays of the angle on any particular circle (see figure at right). So what’s the relationship between degrees and radians? Study the figure to the left. For every angle measured in degrees, there is a related radian measure. The values on the inner ring are

radian measurements. For example, for every 180 degrees there are π radians. Therefore, 180° = π radians. We can use this understanding of the relationship between radians and degrees to go from one to the other using the process of dimensional analysis or unit conversion. While it is possible to use your calculator to transpose radians into degrees and degrees into radians, it is good practice and probably faster and easier to do it with some quick hand calculations as demonstrated next.

radius = x

1 radian(57.3 degrees)



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For example: 120° is equal to how many radians?

radrad3

20811

0211203

2ππ

=°////

×°////

=°

For example: 3π radians equal how many degrees? °=

°=

////

°×

////= 5401

5401801

33dar

darπ

ππ

PRACTICE: Perform the indicated operations. Be sure to simplify your final answers. Convert the following degree measures to radians: 1) °20 2) °50 3) °270

Convert the following radian measures to degrees:

4) 2 radians 5) radians

2π 6) 0 .25 radians

Answers: 1) radians 90π

2) radians 185π

3) radians 2

3π 4) °115 5) °90 6) °14

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