By the end of this lesson, you should be able to:

2. Define of sine, cosine, and tangent in terms of the unit circle.

3. Define cotangent, secant, and cosecant.

4. Use inverse trigonometric functions.

Basic Trigonometry

Geometry of items that exist on a two-dimensional plane is called plane geometry.

SOH CAH TOA

Consider the triangle at the right. It is a right triangle because it has one right (90^o) angle. It has two other angles, and I've labeled one of these Angle M.

We can name the sides or legs of the triangle based on this angle. First, the side of a right triangle that is across from the right angle is called the hypotenuse. Next the side that is adjacent to our Angle M (but not the hypotenuse) we'll call the adjacent side. Finally, the side opposite Angle M we'll label the opposite side.

Trigonometry includes basic functions that allow us to determine the length of any side of a right triangle, given the length of another side and one of the two acute angles. These basic functions are sine, cosine, and tangent. Here is the trick, and you can use the phrase, "SohCahToa," to remember it:
 

SOH	Sine = Opposite / Hypotenuse
CAH	Cosine = Adjacent / Hypotenuse
TOA	Tangent = Opposite / Adjacent

Let's say Angle M measures 55^o and the hypotenuse measures 30 inches, can you figure out how long the adjacent side is, to the nearest tenth of an inch?

Of the three: Soh, Cah, Toa, only Cah mentions both the adjacent and the hypotenuse. Therefore:

Cosine (55^o) = Adjacent / Hypotenuse
and
Adjacent = Hypotenuse * Cosine (55^o)
therefore
Adjacent = 30 inches *.573576
Adjacent = 17.2 inches.

Caution:
If you think your calculator is set to handle angular measurements in degrees, but it is really set for radians, you will get incorrect results. Remember that there are 360 degrees or 2p radians in one circle.

Unit Circle

A "unit circle" is a circle drawn on a Cartesian coordinate plane, with its center at the origin and with a radius of 1. Look at the following figure:

Notice that the circle crosses the x-axis at (1,0) and at (-1,0), and it crosses the y-axis at (0,1) and at (0,-1).

Also, notice that a radius has been drawn from the center to Point A, and that the angle the radius makes with the x-axis is listed as M. In this drawing, M = 20^o. What are the x and y coordinates of Point A?

By definition, the x coordinate is the cosine of M, and the y coordinate is the sine of M. Therefore, the coordinates are (COS(M), SIN(M)) or (COS(20^o), SIN(20^o) or (.9397,.3420).

Also, we can see that for any Point A on the circle, the sine and cosine represent the lengths of two legs of a right triangle, with the radius of the circle as the hypotenuse. Since the radius of the unit circle is 1, we can use the Pythagorean theorem to determine that for any point on the circle, the square of the sine plus the square of the cosine of the angle must equal 1.

Furthermore, the tangent can be defined using the unit circle as the sine divided by the cosine. So, for any angular measurement M, SIN(M) / COS(M) = TAN(M).

Theta:
Typically, the Greek letter, theta (q) is used to represent an angular measurement, not the capital letter M, as was done here.

Cotangent, Secant, Cosecant

The cotangent, secant, and cosecant are the reciprocals of the tangent, cosine, and sine.

Inverse Functions

What is the sine of 22^o? You can find out by using the SIN key on most calculators: .3746. But what angle has a sine of .4542? To determine this, we can use the inverse sine function.

"The angle with a sine of x" is another way of saying "the inverse sine of x" or "the arcsine of x." The arcsine is sometimes indicated by using -1 as an exponent.

Therefore, if you try to find out which angle has a sine of .4542, you could use the SIN^-1 function on many calculators.

If you try this, and the answer is about .47147, then your calculator is set to radians instead of degrees. You can convert to degrees by multiplying an angular measurement in radians by (180^o / p radians). This should give you 27.013^o. Check your answer by finding the sine of 27.013^o and make sure you get .4542.

There are similar functions for the arccosine and arctangent.