## Trigonometry the Easy Way

ב"ה

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

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:

X=10/.5774

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

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:

X=10/.5774

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