00:01
So suppose i have a trig expression here, and i want to try to solve for w.
00:06
Okay, so how can i do that? well, first, notice i have cosine squared and sine squared.
00:10
My first goal here is to write it in terms of one trick function.
00:15
Let's do it in terms of sign.
00:18
Notice that sine squared of an angle plus cosine squared of an angle equals one, right? my pythagorean identity.
00:24
So this means i could solve for cosine squared or get cosine squared to one side.
00:30
So cosine squared is the same thing as 1 minus sine squared than angle.
00:34
So i can replace cosine squared with 1 minus sine squared of w, minus 7 sine of w minus 8 equals 0.
00:45
And let's simplify, let's see what this gives me.
00:47
So distribute the 6 minus 6 sine squared of w, minus 7 sine of w minus 8 equals 0.
00:55
Combined like terms, 6 minus 8.
00:58
This will give me negative 6, sine squared of w.
01:02
Minus 7, sine of w, and 6 minus 8, negative 2.
01:07
So this might be a little hard to see.
01:11
If this is hard to visualize as a quadratic formula, let y equals sign of w.
01:16
And this is negative 6, y squared, minus 7 y, minus 2, 0.
01:21
And just to make this easier, let's multiply both sides to every term by negative 1, 6 y squared plus 7 y plus 2 equals 0.
01:31
And now, let's try to factor this, right? given a quadratic formula, i could try to factor it, or i could use the quadratic equation.
01:40
So my factors of two or two and one.
01:42
And i'm trying to multiply these numbers so that i get, they add to positive seven.
01:47
So i'm going to have positives here.
01:50
Who would my ws be? well, one way to think about this is i'm trying to do six times two is 12, and i'm trying to add them to seven.
02:01
Okay, so what multiplies the 12? it adds seven.
02:04
Well, three and four.
02:05
How do we get three? three times one.
02:08
So my inner terms, and two times two is four.
02:12
So then zero product properties set both equal to zero.
02:19
Solve for w, so strike one from both sides.
02:23
Two, w equals negative one, divided by two.
02:27
W is negative one half...