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|
- 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)
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.
- sin 2x = 2 sin x cos x
cos 2x = cos2 x - sin2 x
- 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.