Trigonometry: A Ninja Turtle's nemesis
I have been a math tutor for several years and I have learned that every student has different strengths and weaknesses. However, I have noticed that a significant proportion of students seem to have trouble dealing with trigonometry. For some students, the whole concept is a bit abstract and as a result, they have trouble working with trig equations and formulas.
Whether you are taking the Math Level I or II SAT subject test, you will need to be familiar with the basics of trigonometry (and in the case of Math Level II, you will need to know a bit more than just the basics), so let’s go over a few important points to keep in mind as you master this topic.
There are three main and very important concepts you need to be very comfortable with:
Sine-it is the ratio of the opposite side (O) of an acute angle in a right triangle to the hypotenuse of that triangle (H). Sin=O/H
Cosine- it is the ratio of the adjacent side (A) of an acute angle in a right triangle to the hypotenuse of that triangle (H). Cos=A/H
Tangent-it is the ratio of the sine to the cosine. Or said another way, the ratio of the opposite side (O) to the adjacent side (A) of the same angle (since the hypotenuse would cancel out in the sine to cosine ratio). Tan=O/A
To get more comfortable with these concepts, you can think of them in relation to the unit circle. In case you are not familiar with this term, the unit circle is a circle, with radius 1 drawn on the coordinate plane. The origin of the plane is also the center of the unit circle. The figure below shows the cosine and sine values of some common angles arranged in the unit circle. Because the hypotenuse is 1 here, the cosine and sine are simply the adjacent (x value of the point on the unit circle) and opposite side (y value of the point on the unit circle) respectively. To simplify matters, when dealing with the unit circle think- sin=y value, cos=x value.
If you pay attention, you will notice that there is a certain pattern:
In Quadrant 1 (angles from 0-90)-both cosine and sine are positive (x and y are positive on the first quadrant). As a result, the tangent (sin/cos) of the angles in this quadrant will also be positive.
In Quadrant 2 (angles from 90-180)- cosine is negative, but sine is still positive (x is negative and y positive in this quadrant). Because sine and cosine have opposite signs, the tangent is negative for angles in this quadrant.
In Quadrant 3 (angles from 180-270)-both cosine and sine are negative (x and y are negative on this quadrant). Since cosine and sine are both negative, the tangent here is positive.
In Quadrant 4 (angles from 270-360)-cosine is positive, but sine is negative (x is positive and y negative in this quadrant). For the same reason as in quadrant 2, the tangent is negative for angles in this quadrant.
While you might have to memorize the sine and cosine values for these common angles, knowing the pattern we just went over should be very helpful. Additionally, you don’t have to memorize the tangent values as you can easily figure them out from the sin to cos ratio. And there is more good news; as you practice over and over with trigonometry problems, you will get to memorize these numbers without even forcing yourself to do so.
Another very important formula you have to know is:
Sin2 q + Cos2 q=1
This makes sense, if you think about it.
Sin=O/H so Sin2 q= O2 / H2
Cos=A/H so Cos2 q= A2 / H2
Sin2 q + Cos2 q= O2 / H2 +A2 / H2 =(A2 + O2 )/ H2
But A2 + O2 = H2 based on the Pythagorean theorem, so
Sin2 q + Cos2 q= O2 / H2 +A2 / H2 =(A2 + O2 )/ H2 =H2 /H2 =1
Realizing that there is a logic behind these complicated-looking formulas and rules can really help students in the course of standardized test prep. They start to realize that this initially-baffling material is not abstract after all and they can actually work through it.
What I have explained here does by no means cover everything you need to know about trigonometry. I have only covered the very basics, the first few bricks needed to build a strong foundation in this area. But everything else rests upon this knowledge, and being confortable with the information covered so far sets you up to excel in the more complicated trigonometry topics. As I said, there are other trigonometric identities and tricks you need to master, and I will go over them another day, in another post. The advice for now: practice makes perfect, so practice trig problems as much as you can. And don’t despair if you find yourself struggling with these problems; talking these problems over with a friend, teacher or math tutor can be very helpful.