The basic trigonometric functions.

If I give you the measure of two interior angles of a triangle, you can figure out the third by adding up the two given numbers and subtracting that number from 180.

In the picture here, the angle on the left is theta degrees and the angle on the lower right is 90 degrees. 180 - (90 + theta) = 90 - theta, which is the measure of the upper right angle.

The two smaller angles must add up to 90 degrees. When two angles have this property, we say the are complementary.  Notice, this is different from complimentary, which means saying something nice. Here are the ways to use complementary in a sentence.

30° and 60° are a pair of complementary angles.

30° is complementary to 60° and vice versa.

That said, the picture above is an example of a class of similar triangles defined by the three angles, 90°, theta° and (90-theta)°. Just as we have not put a number value to theta, we have not put number values to the lengths of the sides, labeled hyp, opp and adj, respective abbreviations for hypotenuse, opposite and adjacent. We could measure theta exactly and give the measure in degrees, but the numbers for the side lengths would depend on what unit we used. Americans would be likely to measure in inches, most of the rest of the world would use centimeters or millimeters, and some clever computer types might try to figure out the number of pixels in each line as drawn in the computer picture.

If the measurements are exact, the numbers produced will be different but the ratios of matching side lengths should be the same, like the hypotenuse divided by the adjacent or the opposite divided by the hypotenuse.  Since we have three numbers, none of which should be zero, we have six different ratios. In trigonometry, there are six basic functions we will associate to an angle measurement, in this case theta.

sine: sin theta = opp/hyp

cosine: cos theta = adj/hyp
tangent: tan theta = opp/adj
cotangent: tan theta = adj/opp (the reciprocal of tangent)
secant: sec theta = hyp/adj (the reciprocal of cosine)
cosecant: csc theta = hyp/opp (the reciprocal of sine)

The prefix "co-" is short for complementary. If we look at the other acute angle (90°-theta), the value for cosine of that angle will be the value for sine of theta, the complementary angle. This is because when we look at the triangle from the point of view of the other angle, the side we had labeled the opposite is now the adjacent and vice versa. The long side is the hypotenuse and it does not change depending on our point of view.

Besides the relationship of complementary values, there are several important trigonometric identities based on The Pythagorean Theorem, which we will discuss tomorrow.

Right triangles: Starting with definitions

Nature does not create many straight lines, with the exception of some crystalline forms. Even rarer in nature are right angles, two straight lines meeting like a plus sign + to create four equal angles that add up to 360°, which means every angle created by the two crossing lines of the plus sign measures 90°.

Straight line geometry has been used for several  millennia by cultures around the world. When we teach geometry today, it is standard to teach the ideas discovered by the Greeks. For example, the symbol over by the angle on the left of this triangle is the lowercase Greek letter theta. It is a common convention to label angles with Greek letters, points with capital letters from the Roman alphabet and the lengths of straight lines with lowercase letters from the Roman alphabet.

In this picture, we have a right triangle. That little rectangle mark in the lower right hand corner signifies a 90° angle.

The side length opposite the right angle is called the hypotenuse. Since the three angles have a sum of 180°, the 90° angle accounts for half of that and the other two angles must add up to 90°. More than that, there is an easily proved property of triangles that the longest side of any triangle will be opposite the largest angle.  Sing the right angle must be the largest angle, the hypotenuse is always the longest side of a right triangle.

The other two sides are called the legs. If we discuss these legs, we often pick one of the smaller angles and distinguish the two legs based on their relation to the chosen smaller angle, which in this case is the one we labeled theta.

The leg that is not on of the line segments next to theta is called the opposite and the leg that creates the angle theta where it intersects with the hypotenuse is called the adjacent.

Tomorrow, we will discuss some of the relationships of the sides and angles of a right triangle using the terms associated with trigonometry.