                  
AC Trigonometry
An electrician does not use trigonometric calculations on the job,
ever. In some thirty years of doing electrical work, I never needed to do a trig
problem for the work. But I needed to know sin cos tan really well for license
exams at various levels. The standard examination for an electrician license at
the Master level has trig as a part of the Alternating Current calculations.
Trig is a way to find missing values in electrical problems where the supply is
alternating current. Thee is no other way to find missing values than to use
trig. So if you have an aversion to trig, and do not plan to learn it for what
ever reason, you'll probably be safe. There are usually fewer than five
questions on any exam where you need to know trig.
On the off chance that you want to learn about this problem solving
technique, here are techniques of solution that will get you through easier than
you would ever think possible. In fact, trigonometry problems, with the aid of a
Texas Instruments TI-30Xa calculator, are no more difficult than substitution
algebra problems.
Admittedly trigonometry can get very difficult. But to the extent that you
need to learn it, right triangle trigonometry, it is not difficult. Dig in.
Trigonometry is important for certain calculations. Here are details about
trigonometry which may help you understand these concepts.
To better understand what the Wright Brothers accomplished and how they
did it, it is necessary to use some mathematical ideas from trigonometry,
the study of triangles. Most people are introduced to trigonometry in high
school, but for the elementary and middle school students, or the
mathematically-challenged:
DON'T PANIC!.
There are many complex parts to trigonometry and we aren't going there. We
are going to limit ourselves to the very basics which are used in the study of
airplanes. If you understand the idea of ratios, one variable divided by
another variable, you should be able to understand this page. It contains
nothing more than definitions. The words are a bit strange, but the ideas are
very powerful as you will see.
Let us begin with some definitions and terminology which we use on this
slide. We start with a right triangle. A right triangle is a three sided
figure with one angle equal to 90 degrees. (A 90 degree angle is called a right
angle.) We define the side of the triangle opposite from the right angle to
be the hypotenuse, "h". It is the longest side of the three
sides of the right triangle. The word "hypotenuse" comes from two
Greek words meaning "to stretch", since this is the longest side.
We pick one of the other two angles and label it b. We don't have to
worry about the other angle because the sum of the angles of a triangle is equal
to 180 degrees. If we know one angle is 90 degrees, and we know the value of b,
we then know that the value of the other angle is 90 - b. There is a side
opposite the angle "b" which we designate o for
"opposite". The remaining side we label a for
"adjacent", since there are two sides of the triangle which form the
angle "b". One is "h" the hypotenuse, and the other is
"a" the adjacent. So the three sides of our triangle are
"o", "a" and "h", with "a" and
"h" forming the angle "b".
We are interested in the relations between the sides and the angles of our
right triangle. While the length of any one side of a right triangle is
completely arbitrary, the ratio of the sides of a right triangle all depend
only on the value of the angle "b". We illustrate this fact at the
bottom of this page.
Let us first define the ratio of the opposite side to the hypotenuse to be
the sine of the angle "b" and give it the symbol sin(b).
sin(b) = o / h
The ratio of the adjacent side to the hypotenuse is called the cosine of the
angle "b" and given the symbol cos(b). It is called
"cosine" because its value is the same as the sine of the other angle
in the triangle which is not the right angle.
cos(b) = a / h
Finally, the ratio of the opposite side to the adjacent side is called the
tangent of the angle "b" is given the symbol tan(b).
tan(b) = o / a
To demonstrate that the value of the sine, cosine and tangent depends on the
angle "b", let's study the three examples at the bottom of the figure.
We are only going to discuss the sin(b) in these examples, but there are similar
discussions for the cosine and tangent. In the first example, we have a 7 foot
ladder that we lean against a wall. The wall is 7 feet high, and we have drawn
white lines on the wall at one foot intervals. The length of the ladder is
fixed. If we incline the ladder so that it touches the 6 foot line, the ladder
forms an angle of nearly 59 degrees with the ground. The ladder, ground, and
wall form a right triangle. The ratio of the height on the wall (o - opposite),
to the length of the ladder (h - hypotenuse), is 6/7, which equals roughly .857.
This ratio is defined to be the sine of b = 59 degrees. (On another
page we show that if the ladder is twice as long (14 feet), and is inclined
at the same angle(59 degrees), that it reaches twice as high (12 feet). The ratio
stays the same for any right triangle with a 59 degree angle.) In the second
example, we incline the 7 foot ladder so that it only reaches the 4 foot line.
As shown on the figure, the ladder is now inclined at a lower angle than in the
first example. The angle is about 35 degrees, and the ratio of the opposite to
the hypotenuse is now 4/7, which equals roughly .571. Decreasing the angle
decreases the sine of the angle. In example three, we incline the 7 foot ladder
so that it only reaches the 2 foot line, the angle decreases to about 17 degrees
and the ratio is 2/7, which is about .286. As you can see, for every angle,
there is a unique point on the wall that the 7 foot ladder touches, and it is
the same point every time we set the ladder to that angle. Mathematicians call
this situation a function.
Since the sine, cosine, and tangent are all functions of the angle
"b", we can determine (measure) the ratios once and produce tables of
the values of the sine, cosine, and tangent for various values of "b".
Later, if we know the value of an angle in a right triangle, the tables tells us
the ratio of the sides of the triangle. If we know the length of any one side,
we can solve for the length of the other sides. Or if we know the ratio of any
two sides of a right triangle, we can find the value of the angle between the
sides. We can use the tables to solve problems. Some examples of problems
involving triangles and angles include the descent
of a glider, the torque
on a hinge, the operation of the Wright brothers' lift
and drag balances,
and determining the lift to
drag ratio for an aircraft.
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