Trig. Identities

The trigonometric fuctions that were introduced in the previous sections are all related to each other through different equations. These are called the TRIGONOMETRIC IDENTITIES.

We will first take a look at the Reciprocal Identities:

csc t = 1/sin t sec t = 1/cos t cot t = 1/tan t
tan t = sin t /cos t cot t = cos t /sin t
These are easy to prove just by using the definitions of each of the trig. functions. But in case you get stuck, here are the proofs.

sin2 t = (sin t)2 = (sin t)(sin t).
sink t = (sin t)k
This is true for all trigonometric functions.

Next, we will look at the Pythagorean Identities. Recall that we can place a right triangle in the unit circle like so:

Then we get the triangle with the sides and hypotnuse as follows:

Using Pythagorean's Thoerem, we get sin2 t + cos2 t = 1

Now dividing by sin2 t we end up with 1 + cot2 t = csc2 t.
By doing a similar thing with cos t the result is tan2 t + 1 = sec2 t

Here are a few examples using these identity properties.

Another important set of identities are the addition and subtraction formulas. These are only for sine and cosine and are as follows.

    sin (s+t) = (sin s)(cos t) + (cos s)(sin t)

    sin (s-t) = (sin s)(cos t) - (cos s)(sin t)

    cos (s+t) = (cos s)(cos t) - (sin s)(sin t)

    cos (s-t) = (cos s)(cos t) + (sin s)(sin t)

Why would we need such formulas? Here are some examples of where this would be needed.

The last topic for this section is Double and Half Angle Formulas. These are just as useful as the above addition and subtraction formulas. The double-angle fromula allow us to find the values of the trig. functions at 2x, and the half-angle formulas relate the values of the trig functions at x/2 to their values at x.

Double-angle Formulas:

    sin 2x = 2 sin x cos x
    cos 2x = cos2 x - sin2 x
These are gotten by the addition formulas replacing both s and t by x. The half-angle formulas are based on the double-angle formulas and are given by:
    sin2 = (1 - cos 2x)/2
    cos2 = (1 + cos 2x)/2

Let's do a couple of examples to practice this idea before moving to the Exercises.

Introductory Exercises Moderate Exercises