Lets look at trig values give information of the angle.

For example, Let (12, -5) be a point on the terminal side of theta.

Find the exact values of cos theta, csc theta, and tan theta.

Now, the first thing to notice is that this point here, (12, -5) does not lie on

the unit circle, because in order for a point to lie on the unit circle, x^2 +

y^2 has to be equal to 1. But looking here, we have 12^2 + (-5)^2 =

144 + 25, which is equal to 169, which is not 1.

Therefore, we cannot use the unit circled definition of the trig functions, but

there is an equivalent one. [SOUND] That is, if x, y is the point of

intersection of the terminal side of the angle and a circle or radius r, then the

sine of the theta is equal to y / r, cosine theta is equal x /. r, tangent of

theta is y / x, cosecant of theta is r / y, secant of theta is r / x, and

cotangent of theta is x / y. And we're assuming here that r > 0 or

that x, y is not the origin. So, in our case, we have that (x, y) =

(12, -5) [SOUND] as shown in this figure here.

So we'll find r by using this formula here, namely r is equal to the square

root of x squared, which is 12^2 + -5^2 or the square root of 144 + 25,

which is equal to the square root of 169, which is equal to 13.

So let's write that on the graph up here, r = 13,

and now that we know r, we can find these three values here by using this equation,

this equation, and this equation.

That is cos theta is equal to x / r which is equal to 12 / 13, cosec theta is equal

to r / y which is equal to 13 / -5 or -13 / 5, and tangent of theta is equal to y /

x or -5 divided by 12 which is -5/12. Let's look at another example.

[SOUND] Let -3, square root of 7 be a point on the terminal side of theta.

Let's find the exact values of sine of theta, secant of theta, and and cotangent

of theta. Again, our given point here does not lie

on the unit circle, because -3^2 plus the square root of 7^2

= 9 + 7, which is 16, which is not equal to 1.

Therefore, we cannot use the unit circle definitions of the trig functions.

[SOUND] So we're going to use these definitions instead,

and here, we have that (x, y) = -3, square root of 7 [SOUND] as shown in this

figure here. So let's find r by using this equation

again, namely r is equal to the square root of

x^2 or -3^2 plus the square root of 7^2, which is equal to the square root of 9 +

7, which is equal to the square root of 16

or 4. So let's write that here in the figure.

This is equal to four. Now, notice in both of these examples,

we're basically doing the same work twice,

because looking up here, didn't we already know that r^2 was 16? So

basically, you would just have to take the square root of that number to get

your r each time. Alright, now, we're ready to find the

sine of theta, secant of theta, and cotangent of theta by this formula, this

formula, and this formula. That is sine of theta is equal to y / r,