We review trigonometric functions.

### What are trigonometric functions?

**trigonometric function**is a function that relates a measure of an angle of a right triangle to a ratio of the triangle’s sides.

The basic trigonometric functions are cosine and sine. They are called “trigonometric” because they relate measures of angles to measurements of triangles. Given a right triangle

At this point we could simply assume that whenever we draw a triangle for computing sine and cosine, that the hypotenuse will be . We can do this because we are simply scaling the triangle, and as we see above, this makes absolutely no difference when computing sine and cosine. Hence, when the hypotenuse is , we find that a convenient way to think about sine and cosine is via the unit circle:

If we consider all possible combinations of ratios of

adjacent, opposite, hypotenuse,

(allowing the adjacent and opposite to be negative, as on the unit circle) we obtain all of the trigonometric functions.

### Not all angles come from triangles.

Given a right triangle like

*other*angles, we need to be able to describe the trigonometric functions a little more generally. To do this, we use the unit circle from the previous section. Given an angle , we construct the angle with initial side along the positive -axis and vertex at the origin.

### Graphs

As a reminder, we include the graphs here.

### The power of the Pythagorean Theorem

The Pythagorean Theorem is probably the most famous theorem in all of mathematics.

The Pythagorean Theorem gives several key trigonometric identities.

There several other trigonometric identities that appear on occasion.

If we plug into the angle addition formulas, we find the double-angle identities.

Solving the bottom two formulas for and gives the half-angle identities.

### Trigonometric equations

Frequently we are in the situation of having to determine precisely which angles satisfy a particular equation. The most basic example is probably like this one.

To find all solutions, we have to add all multiples of to these. The solutions are then

Let’s try one a bit more complicated.

Notice that this equation is quadratic in . We can factor it like we try to do to solve any other quadratic equation: On the interval , has only one solution, . For , we see that the reference angle . Since cosine is positive in quadrants 1 and 4, we find solutions and .

All solutions are:

### Limits involving trigonometric functions

Back when we introduced continuity we mentioned that each trigonometric function is continuous on its domain.

We’ll end with a couple very involved limits where the Squeeze Theorem makes a surprising return.

From our diagrams above we see that and computing these areas we find Multiplying through by , and recalling that we obtain Dividing through by and taking the reciprocals (reversing the inequalities), we find Note, and , so these inequalities hold for all . Additionally, we know and so we conclude by the Squeeze Theorem, .

When solving a problem with the Squeeze Theorem, one must write a sort of mathematical poem. You have to tell your friendly reader exactly which functions you are using to “squeeze-out” your limit.