Trigonometry Review

Trigonometry is a vital part of mathematics, and is used extensively in all of the physical sciences, including physics per se, and its subsidiary subject, astronomy.Trigonometry comes from geometry, specifically a series of geometric theorems regarding similar right triangles.Right triangles are triangles that have a 90 degree or right angle therein. Two right triangles are similar if they share the same angles within them. We say the triangles are similar (and not identical, or congruent), because in general, one triangle is bigger than the other.Let's look at one of the angles of a right triangle other than the 90 degree angle. Call this angle the angle of interest. In the screenshot above, that angle is indicated by theta. Its complementary angle (90 degrees minus theta) is indicated by theta sub c.The side of the triangle opposite of theta has a length equal to "a". The side adjacent to (or part of) theta that is not the longest side has a length of "b". The longest side of the triangle, the hypotenuse, is opposite the 90 degree angle, and has a length of "c".

Geometry teaches us, and trigonometry algebraically formalizes, the following statement:

Ratios of sides of similar right triangles are equal.

What this means is that the value of any ratio of triangle side lengths, say "a" divided by "c", is independent of the triangle per se, and is only a function of the angle theta.

We define the three main trigonometric ratios as follows: a/c is the sine of theta, b/c is the cosine of theta, and a/b is the tangent of theta. A cursory observation of any right triangle reveals that the sine of a given angle (theta) is equal to the cosine of its complementary angle (theta sub c).

Many people use the mnemonic "SOHCAHTOA" to remember the side length relationships that define sine (sin), cosine (cos) and tangent (tan). The mnemonic means "sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent".

In this animation (corresponding to the screenshot above), you can change both the angles (theta and theta sub c), and the size of the triangle, by dragging on the black dot.

If you press the button "P" on the animation, you will switch to the "pythagorean" view of the animation. Here we display how "the sum of the squares of the sides of a right triangle equal the square of the hypotenuse".Again, you can change both the angles (theta and theta sub c), and the size of the triangle, by dragging on the black dot.It is left as an exercise for the student to show that the square of an angle's sine added to the square of an angle's cosine must add up to one.Pressing "P" again returns you to the original view of the animation.

Experiment

The original view of the animation defaults to a starting value of theta equal to 30 degrees. It is easy to show that the sine of 30 degrees is one half, or 0.5, and that the cosine of 30 degrees is 0.866 (equal to the square root of 3 divided by 2). Notice that the sine of 30 equals the cosine of 60 and the cosine of 30 equals the sine of 60, since 30 and 60 degrees are complementary angles.

Now drag on the black dot so as to not change the value of theta. Notice that the lengths a, b and c all change, but that the sine, cosine and tangent of theta do not: this consistency illustrates that sine, cosine and tangent are properties of the angle theta, and are not peculiar to a specifically sized right triangle.

Now drag the black dot so as to change theta to 60 degrees. You should see the following results: sine of 60 degrees is 0.866, and cosine of 60 degrees is 0.5. Did you expect these results?

Now select an arbitrary angle (say 53 degrees). Write down the values of the angle's sine, cosine and tangent. Is the ratio of the angle's sine divided by its cosine equal to its tangent? Is so, why?

Try one more value of theta, namely 45 degrees. What do you get for its sine, cosine and tangent?

At this point, you should make sure that you can perform trigonometric operations on your calculator.

Now press the "P" button. For any angle selected, is "a squared plus b squared equal to c squared"? Do the squares of an angle's sine and cosine add up to 1 (within a rounding error)?

For a famous example of getting the Pythagorean Theorem wrong, see this movie clip.