Wednesday, February 26, 2014

What in heck is "Soh" "Cah" "Toa"?

Trigonometry the Easy Way


Many of my tutoring students have trouble with trigonometry. But the main part of trig is quite simple if you know some tricks. All you have to do is keep a few things in mind and you can solve just about any trig problem.

Sines, Cosines and Tangents

Students who are new to Trigonometry have some difficulty with the concept. Basically, what "trig" says is that, given a right triangle, there are certain ratios of the sides to each other based on the measure of the angles that are not right angles. What this mean is that, if a right triangle has, for example, an angle of 30 degrees, the ratio of the length of the side opposite the 30 degree angle and the hypotenuse (the side opposite the right angle) -- called the "sine" -- will always be the same.

The basics of a "trig" problem is that, given two pieces of information, you can always find the third. The three pieces of information are the length of two sides of the triangle and the angle measurement of one of the angles.

(Each of these functions is as it relates to a specific angle. "Opposite" is the side of the triangle that is opposite the angle, "Hypotenuse" is the side opposite the right angle and "Adjacent" is the other side. So, when speaking of the two acute -- that is under 90 degrees -- angles in a right triangle, what is "opposite" to one is "adjacent" to the other. For example, in the triangle pictured here, for the 30 degree angle, the side labeled "Opposite" is "Adjacent" to the other angle -- the 60 degree angle which is created where the "Hypotenuse" and the side labeled "Opposite" come together at the top of the triangle -- and the side labeled "Adjacent" is adjacent to the 30 degree angle but opposite when referring to the 60 degree angle.)

Because Sine is "opposite/hypotenuse" and Cosine is "adjacent/hypotenuse" and Tangent is "opposite/adjacent" , there is a mnemonic to remember this -- Soh Cah Toa .

If the side opposite the 30 degree angle is 10 and you want to know the adjacent side's length, what you need to do is find out the Tangent (tangent -- tan -- is opposite over adjacent -- opp/adj) and set up the problem with what you know:

Tan 30 degrees = 10/X (we know the opposite is 10, we don't know the adjacent side, so we assign "X" -- a variable, meaning it's unknown)

You then look up the tangent of 30 degrees (on a trig table or using your calculator) and you can find out that the tangent of 30 degrees is .5774 so you can set it up as:

.5774 = 10/X

Move the "X" to the other side by multiplying both sides by "X":

.5774X = 10

Isolate the "X" by dividing each side by .5774:


Which comes out to approximately 17.3 -- this is what the length of the adjacent side would be.

Algebra: Exponents

Explaining Exponents

In the years I've been tutoring, I have noticed that many people find the concepts of exponents to be a very daunting one. But exponents are really only an easier, more understandable way of expressing larger, more cumbersome numbers. I hope you feel less intimidating after reading this.

Firstly, Some Simple Exponent Rules

Before you check out the rest of this lens, keep 2 things in mind:

1) You cannot add numbers with exponents UNLESS they have the same exponent and base AND

2) The base (that is the number being raised to the power) needs to be the same to be able to multiply or divide them.

For example, you cannot add the terms in figure 1 because they have different exponents. You cannot simplify the terms in figure 2 because they have different bases (a, b and c).

Explaining Exponents

An exponent means that's how many times you multiply a number by itself. As you can see from the picture, A squared (that's A to the 2nd power) is A times A. A cubed (A to the 3rd power) is A times A times A. A to the 4th power is A times A times A times A.

Multiplying Exponents

Multiplying numbers with exponents is relatively easy -- all you do is add the exponent.

If you look at the graphic, you can see why that works -- you are adding the "A"s in the multiplication problem. The "A" squared represents two "A"s multiplied by each other, the "A" cubed represents three "A"s multiplied by each other. So you add the "A"s and you get "A" to the fifth power.

Dividing Exponents

Dividing exponents is pretty much just as easy as multiplying, but reversed. To divide numbers with exponents, you subtract the exponents. As you can see from this graphic, "A" cubed divided by "A" squared is "A" to the first, or "A".

In order to raise an exponent to another power, you multiply the exponents. As you can see from the graphic, each "A" Squared appears three times -- so if you count the "A"s you will see there is a total of 6 "A"s.

Negative Exponents

A negative exponent is just 1 over the same number exponent -- just as multiplying has you adding exponents, using negative exponents would have you add the exponents --

Scientific Notation

Scientific Notation is the term for using the powers of 10 (each of which represent one zero) to express complex numbers. For example, in the graphic as the right, 406,000,000 is difficult to deal with. So if you express this as 4.06 X 10 to the 8th power, then it is easier to work with. On the other hand, .000000406 is 4.06 X 10 to the -7 (that is 1/10 to the seventh power).

10 to the 8th power is a 1 with 8 zeroes, 100,000,000 and 4.06 times that means that the 4 would be where the 1 is in 100,000,000 and the .06 would follow the 4 into the next two spots in the number.