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We review trigonometric functions.

### What are trigonometric functions?

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

we define Note, the values of sine and cosine do not depend on the scale of the triangle. Being very explicit, if we scale a triangle by a scale factor $k$, and

At this point we could simply assume that whenever we draw a triangle for computing sine and cosine, that the hypotenuse will be $1$. 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 $1$, 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

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

Which of the following expressions are equal to $\sec (\theta )$?
$\frac {1}{\cos (\theta )}$ $\frac {1}{\sin (\theta )}$ $\frac {\text {adj}}{\text {hyp}}$ $\frac {\text {hyp}}{\text {adj}}$ $\frac {\tan (\theta )}{\sin (\theta )}$ $\frac {1}{\sin (\theta )\cdot \cot (\theta )}$

### Not all angles come from triangles.

Given a right triangle like

the angle $\theta$ cannot exceed $\frac {\pi }{2}$ radians. That means to talk about trigonometric functions for 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 $\theta$, we construct the angle with initial side along the positive $x$-axis and vertex at the origin. As the angle $\theta$ grows larger and larger, the terminal side of that angle spins around the circle. The trigonometric functions of the angle $\theta$ are defined in terms of the terminal side. From the picture, you see that this agrees with what you know about trigonometry for triangles, but it allows us to extend the definition of sine and cosine to all real numbers, instead of only the interval $\left ( 0, \frac {\pi }{2} \right )$

### 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 $s = t = \theta$ into the angle addition formulas, we find the double-angle identities.

Solving the bottom two formulas for $\cos ^2(\theta )$ and $\sin ^2(\theta )$ 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.

Solve the equation:
$\theta = \frac {\pi }{3}+\pi k$ $\theta = \frac {\pi }{6}+ \pi k$ $\theta = \frac {2\pi }{3}+ \pi k$ $\theta = \frac {5\pi }{6}+\pi k$ None of the above

Let’s try one a bit more complicated.

### Limits involving trigonometric functions

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

Compute the limit:

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

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.