
Basic Functions
Most of the trigonometry on the Math IIC test has to do
with the different parts of a right triangle and the relationships
among these different parts. The three basic trigonometric functions—sine,
cosine, and tangent—are the tools that define these connections.
Given the measure of one of the nonright angles in a right triangle,
you can use these tools of trigonometry to find the shape of the
triangle. If you are given the measure of one of the nonright angles
and one of the sides, you can find all the values of the right triangle.
Basic Functions and the Right Triangle
If you know the measure of one nonright angle, the trigonometric
functions tell you the ratio of the lengths of any two sides of
the right triangle.
In the right triangle below, the measure of one acute
angle is labeled , and the sides
of the triangle are labeled hypotenuse, opposite, and adjacent,
according to their position relative to the angle of measure .
Sine
The sine of an angle is the ratio of the side opposite
the angle over the hypotenuse.
Cosine
The cosine of an angle is the ratio of the side adjacent
the angle over the hypotenuse.
Tangent
The sine of an angle is the ratio of the side opposite
the angle over the side adjacent to the angle.
A handy way to remember these formulas is the acronym
SOHCAHTOA. The “S,” “C,” and “T” stand for the three different basic
trigonometric functions, while the two letters after the “S,” “C,”
and “T” refer to the sides of the triangle that are being related
by that function.
 SOH: Sine is the side Opposite the angle divided by the Hypotenuse.
 CAH: Cosine is the side Adjacent to the angle divided by the Hypotenuse.
 TOA: Tangent is the side Opposite divided by the Adjacent side.
Using Your Calculator with the Basic Functions
On some questions dealing with sine, cosine, and tangent,
your calculator can be extremely helpful. Using your calculator,
you can quickly compute the value of one of the three trigonometric
functions at any given angle. On a graphing calculator, you would
find the button indicating the trigonometric function you want to
perform, type in the value of the angle, and then hit Enter. To
calculate the cosine of 45º, press the COS button, then type in 45
and press Enter.
On nongraphing calculators you may need to type in the
value of the angle first and then press the trigonometric function
button.
Angles Larger Than 90º and the Basic Functions
Angles in a right triangle can never be larger than 90º,
since the sum of all three angles must equal 180º. But on the Math
IIC you may occasionally run into angles that are larger than 90º.
It is often more intuitive to think of these in terms of the coordinate
plane rather than in terms of a triangle.
Below are pictured four angles in the coordinate plane.
The first is the acute angle we’ve already covered in this chapter;
the next three are all larger than 90º.
The four quadrants of the coordinate plane become very
important when dealing with angles that are larger than 90º. Each
angle larger than 90º can be “simplified” by looking at it in the
context of its own quadrant. In the figure below, the four angles
from the previous angle are defined in terms of their own quadrants:
By reconsidering each angle based on its relationship
to the xaxis, it becomes clear that each of the
original angles can be treated as a reoriented 30º angle. In other
words, a 210º angle is just the same as a 30º angle except that
the 210º angle lives in the third quadrant. In terms of the basic
trigonometric functions, this means that the value of a 210º angle
is the same as the value of a 30º value, except that the sign of
the trigonometric function differs based on the quadrant in which
the angle exists. Depending on the quadrant of the coordinate plane
in which an angle resides, the values of the trigonometric properties
of that angle will be either positive or negative. Below is a figure
illustrating the signs of the trigonometric functions according
to the quadrant in which they lie.
You should memorize this chart.
The Math IIC will probably test whether you know the proper
sign for each quadrant in an indirect way, meaning that it’s unlikely
that you’ll have to do any heavy calculating when dealing with this
topic. Instead, you might find a question such as:

This question doesn’t ask you to think about sine for
any specific value of . Instead, it tests
your understanding of the quadrant signs for the sine function.
The first thing you should see is that and have the same magnitude, even if
they have different signs. This means that the magnitude of sine
for and will be the same. Immediately you
should understand that sin must equal either 0.5 or –0.5. To
figure out which of these values is right, you have to decide what
quadrant angle resides in. Based
on the graph of the sine function or from the above chart, you can
see that the sine function has a positive value in Quadrants I and
II, and negative values in Quadrants III and IV. Since sin is equal to a positive number, .5,
you know that must represent
an angle in Quadrant I or II. Since angle is simply the reflection of across the xaxis,
you can see that angle must be in either
Quadrant III or IV. The value of sin must be negative: –.5 is the right
answer.
