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# Trigonometry: Complete Set of Lessons

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T01-Angles.mws

High School Modules > Trigonometry by Gregory A. Moore

Angles

Exploration of Degree and Radian measure of angles.

[Directions : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in this worksheet.]

0. Code

> restart; with(plots): with(plottools):

```Warning, the name changecoords has been redefined
```

```Warning, the name arrow has been redefined
```

> libname:="C:\\Program Files\\Maple 7\\LIB\\Trig\\", libname:

1. Degrees

A degree is 1/360 of a rotation. If you were to stand in one point and rotate, your view would sweep out 360 degrees in a complete rotation. One degree, is therefore, very small.

> DegreePlot(1);

30 degrees is a more reasonably sized angled.

> DegreePlot(30);

90 degrees is called a "right angle".

> DegreePlot(90);

135 degrees is into the second quadrant.

> DegreePlot(135);

Above 180 degrees, angles start to get into the 3rd quadarant.

> DegreePlot(210);

And above 270 degrees, angles get into the 4th quadrant.

> DegreePlot(300);

Why is 360 used? The ancient Babylonians computed there were 360 days in a year. The passage of seasons is a universally known cycle, so the Babylonians assigned degrees of a circle to days of the year. 360 is also a convenient number to be divided! Since we often deal with fractions of a circle, its convenient to divide 360 by various denominators. Here is the set of 24 whole numbers that divide 360. In fact, the smaller integer that does NOT divide 360 is 7.

> numtheory[divisors](360);

> for k from 1 to 23 do cat(numtheory[divisors](360)[k],` * `,
360/numtheory[divisors](360)[k], ` = 360`); od;

So 1/2 of a circle has a degree measure of 1/2 of 360.

> 360/2; DegreePlot(%);

So 1/4 of a circle has a degree measure of 1/4 of 360.

> 360/4; DegreePlot(%);

So 1/3 of a circle has a degree measure of 1/3 of 360.

> 360/3; DegreePlot(%);

So 1/8 of a circle has a degree measure of 1/8 of 360.

> 360/8; DegreePlot(%);

So 1/12 of a circle has a degree measure of 1/12 of 360.

> 360/12; DegreePlot(%);

2. Minutes & Seconds

Each degree can be further broken into 60 minutes, and each minute can be broken in 60 seconds. Fractional degrees can be represented in this system. You can think of a giant clock with 360 hours, 60 minutes per hour, and 60 seconds per minute.

Convert : Degrees/Minutes/Seconds to Decimal Degrees

Example 2.1 : Express 34 degrees, 42 minutes, 29 seconds in decimal form
.

> 34 + 42/60 + 29/60^2;

>

Example 2.2 : Express 119 degrees, 7 minutes, 57 seconds in decimal form
.

> 119 + 7/60 + 57/60^2;
evalf(%, 15);

Convert : Fractional/Decimal Degrees to Degrees/Minutes/Seconds

Example 2.3 :
Express 72.8921 degrees in Degree/Minute/Second form
.

> 72.8921;
a := evalf(%);

> degrees := floor(a);

> fractional_degree := a - degrees;

> minutes := floor( 60*fractional_degree );

> fractional_min := fractional_degree - minutes/60;

> seconds :=floor( 60^2 * fractional_min);

> `Compare & Check`;
evalf(a);
print(cat(degrees, ` degrees, `, minutes, ` minutes, `, seconds, ` seconds`));
evalf( degrees + minutes/60 + seconds/60^2);

Note that these two values are slightly different. This is because we rounded off the seconds. If we left the decimals to be a decimal number too, then these both would agree.

> seconds := 60^2 * fractional_min ;

> `Compare & Check`;
evalf(a);
print(cat(degrees, ` degrees, `, minutes, ` minutes, `, seconds, ` seconds`));
evalf( degrees + minutes/60 + seconds/60^2);

Example 2.4 : Express 18 5/7 degrees in Degree/Minute/Second form
.

> 18 + 5/7;
a := evalf(%);

> degrees := floor(a);

> fractional_degree := a - degrees;

> minutes := floor( 60*fractional_degree );

> fractional_min := fractional_degree - minutes/60;

> seconds :=floor( 60^2 * fractional_min);

> `Compare & Check`;
evalf(a);
print(cat(degrees, ` degrees, `, minutes, ` minutes, `, seconds, ` seconds`));
evalf( degrees + minutes/60 + seconds/60^2);

Another way of measuring angles is to compute the arc length that the angle corresponds to on a unit circle. Since the circumference of an entire unit circle is
, that is the radian measure of a complete rotation. Therefore, fractions of a circle correspond to fractions of radians.

1/2 of a circle is 1/2 of

> (2*Pi)/2; AnglePlot(%);

1/4 of a circle is 1/4 of

> (2*Pi)/4; AnglePlot(%);

1/8 of a circle is 1/8 of

> (2*Pi)/8; AnglePlot(%);

1/12 of a circle is 1/12 of

> (2*Pi)/12; AnglePlot(%);

So what does 1 radian look like? (We'll find out how many degrees this is later.)

> AnglePlot(1);

The key to conversion between the two types of angle measuring systems is that half of circle is 180 degrees and

Convert : Radians to Degrees : Multiply by

> Pi/4; % * 180/Pi;

Note that 45 degrees and radians represent the same angle

> DegreePlot(45);

> AnglePlot(Pi/4);

Here are other examples

> 2*Pi/3; % * 180/Pi;

> Pi/6; % * 180/Pi;

> 7*Pi/6; % * 180/Pi;

> 5*Pi/3; % * 180/Pi;

> 11*Pi/6; % * 180/Pi;

There is also the built-in Maple function to make these conversions.

Convert : Degrees to Radians : Multiply by

> 60; % * Pi/180;

> 135; % * Pi/180;

> 240; % * Pi/180;

> 315; % * Pi/180;

> 330; % * Pi/180;

There is also the built-in Maple function to make these conversions - note to "multiply" the number of degrees by the word "degrees."

> with(Units[Standard]):

```Warning, the assigned name polar now has a global binding
```

```Warning, these protected names have been redefined and unprotected: *, +, -, /, <, <=, <>, =, Im, Re, ^, abs, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, argument, ceil, collect, combine, conjugate, convert, cos, cosh, cot, coth, csc, csch, csgn, diff, eval, evalc, evalr, exp, expand, factor, floor, frac, int, ln, log, log10, max, min, normal, root, round, sec, sech, shake, signum, simplify, sin, sinh, sqrt, surd, tan, tanh, trunc, type, verify
```

5. Famous Angles

There are various angles which are used frequently in geometry and trignometry - in particular the multiples of 30, 45, 60, and 90 degrees - or in radians
. It's a good idea to become familiar with all multiples of these angles in both degrees and radians.

> AngleSpectrum(Pi/2);

> AngleSpectrum(Pi/4);

> AngleSpectrum(Pi/8);

> AngleSpectrum(Pi/6);

> AngleSpectrum(Pi/12);

> AngleSpectrum(Pi/3);

6. Angles & Arclength

There is another advantage of using radians when it comes to finding the length of an arc. Since a radian measure is already an arclength on a unit circle, any larger or smaller circle can be scaled appropriately - in a since all circles are "similar" circles to the unit circle. So the arclength of an arc with central angle theta is