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HP 30S Converting Angles and Times

Angle Measurements Time Measurements Practice Solving Problems Involving Angles and Times hp calculators HP 30S Converting Angles and Times

Angle measurements

The amount of turning between two rays with a common origin, or the measure of change of direction, is what we call an angle. Angles cannot be measured in meters or yards. As measures of the amount of turning, we use degrees, radians and grads.

A degree is one 360th of a complete turn and its symbol is º. Just like a meter which has multiples and submultiples (e.g. kilometer, millimeter, etc), degrees are broken down into minutes and seconds. The minute (‘) is one sixtieth of a degree, and the second (‘‘) is one sixtieth of a minute. The radian is defined as the angle subtended by an arc of a circle of length equal to its radius (see figure 1). Its symbol is C (but this symbol is not widely used). Note that the counterclockwise direction of turning is taken to be positive. Grads are a measure of angle equal to one hundredth of a right angle. Grads are also known as gradians and grades. Angle units can be easily converted since:

It is important to understand that there are two ways of expressing an angle in degrees: using decimal degrees and using degrees, minutes and seconds. In decimal degrees, an angle is simply given as 3.5 degrees, that is to say, 3 degrees and a half. In the degrees, minutes, seconds (or DMS) format, the same angle is 3 degrees, 30 minutes.

The HP 30S can work with angles in either measurement system and provides the DRG ( c) and DMS ( —A) menus to convert between them. It also has the DMS function (—A<y) which is useful in converting between the two formats of angles expressed in degrees.

Time measurements

As units of time, a minute is equal to 60 seconds and a second (abbreviated as s) is the fundamental unit of time in the SI system. Since an hour has 60 minutes, a useful application of the conversion between decimal degrees and DMS angles is that the exact same conversion can also work for time. A measurement of 10.5 hours can be converted into 10 hours and 30 minutes by the same process by which an angle of 10.5 degrees can be converted into 10 degrees, 30 minutes (i.e. using the same function, DMS)

Practice solving problems involving angles and times Example 1: Convert an angle of 25 degrees into radians.

Solution: Like many other calculators, the HP 30S operates in an angle mode that the user should specify before doing certain calculations such as trigonometric functions and coordinate conversions. Such mode can be selected in the DRG menu. Press c and use the < and @ keys to select the desired mode, in this case, select RAD and press y. The RAD annunciator is lit . Now , all angles will be in radians unless stated otherwise. We need to enter 25 degrees, so we must find a way of telling the calculator that this angle is not expressed in radians. This is done by placing the degree symbol (º) after the quantity 25, and such symbol is in the DMS menu. Press:

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The entry line contains 25º. When this is evaluated by the ykey, the HP 30S will convert 25º into the current angle unit, i.e. radians.

Answer: 0.436332313 radians. Example 2: Convert 25º into grads. Solution: First of all, we have to set the angle mode to the unit into which we want to convert, i.e. grads. Press c@y (assuming RAD was the current mode)

Note that the entry line now contains 25º again—the last calculation entered or the one being edited— so we just have to press y to do the conversion.

Answer: 27.77777778 grads or 9727 grads (press —Iy to express the result as a mixed number).

Example 3: Convert an angle of 3 2π radians into decimal degrees.

Solution: Since the target unit is degrees, set the angle mode to degrees by pressing: c@y. The entry line still contains 25º from previous examples. Press o to clear it. Now enter 32π and add a small r from the DMS menu to indicate that this quantity is expressed in radians:

r2x/3s—A@@@y

The entry line now reads r)(32π. Parentheses are necessary because the º, ‘, ‘’, and functions take precedence over the division function. Press yand the result line will contain the same value but expressed in decimal degrees.

r g

Answer: 120º

Example 4: Add 120º to and express the result in both decimal degrees and in the DMS format. r.14 Solution: 120º is already stored in ANS from the previous example, so press: +4.1—A@@@y

It returns the sum expressed in decimal degrees. To express this number in the DMS format, use the

DMS function of the DMS menu. You need not reenter its argument, since it is taken from the ANS register, simply press:

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Example 5: Convert an angle of 2.3714 degrees into DMS. Solution: Assuming that the current angle mode is still set to degrees, press: 2.3714—A<y

Answer: 2º 2’ 17.04’’ Note that seconds are expressed in decimal format. Example 6: Convert an angle of 0.43 radians into DMS. Solution: You don’t have to convert 0.43 radians into decimal degrees first. Instead, use the DMS function directly: .43—A@@@y—A<y

Answer: 24º 38’ 13.9‘’ Example 7: Convert an angle of 118º 27’ 59’’ into decimal degrees.

Solution: On the HP 30S there is no specific function to convert DMS into decimal degrees, because such conversion is done automatically when processing the entry line, i.e. after pressing the y key. The º, ‘ and ‘’ symbols are in the DMS menu:

118—Ay27—A@y59—A@@y In fact, the first two yare not necessary – selecting the desired symbol in the DMS menu suffices: 118—A27—A@59—A@@y The result line displays the angle in decimal degrees.

Answer: 118.4663889º’ Example 8: Add 5 hours 3 minutes to 3 hours 58 minutes.

Solution: Remember that time measurements can be calculated using angle functions. This calculation can be done thus: DMS: )'º'º(583335+ r5—A33—A@+3—A58—A@ s— A<y

Answer: 9 hours and 31 minutes.

Note. More than one or (or both) can be included in the same calculation. Its action depends on the current

angular unit. For example, in grad mode is the same as . It does not work for º because degrees are parsed differently because of the DMS format.

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