00:01
In this video, we are going to evaluate this expression that we see here.
00:04
We're dealing with a sine function with the difference of two different angles, and that tells us the difference formula that we see below here can be used.
00:15
Let's start off by writing by that formula, sine of the first angle, sine inverse of negative four -fifths, times cosine of the second angle, which is tangent inverse of three -fourths, and subtract from that cosine of the first angle which is sine inverse of negative four -fifths times sign of the first or second angle which is tangent inverse of three -fourths so that's our setup with a difference formula for sign when we look at this layout we see there is one cancellation sign of sine inverse cancels and we get 4 .5s.
01:05
So we have three evaluations to work on.
01:08
Let's write down on our next step that we have negative four -fifths times the result of evaluating cosine of tangent inverse of three -fourths.
01:20
Let's do that evaluation down below.
01:22
So we want to know what is cosine of tangent inverse of three -fourths.
01:31
To determine that evaluation, we make a substitution.
01:35
Let theta be equal to tangent inverse of three -fourths.
01:42
Then we have cosine of theta that we are evaluating by substitution, and once we obtain a value here, we'll be ready to plug in that value there from the last equation.
01:56
So to do this evaluation, we can take tangent on both sides to obtain tangent theta equals three -fourths, where tangent of tangent inverse cancels out.
02:07
Now we can use a pythagrin identity that says tangent squared of theta plus 1 is equal to secant squared of theta.
02:19
Our tangent itself is three -fourths, so when we substitute it here and square it, we get 916 plus 1, which is 1616 equals secant squared of theta.
02:34
So in other words we now have 2516th is secant squared of theta, and upon taking the root of both sides, we have the positive root of 5 fourths is secant theta.
02:50
Well, that seems kind of frustrating because we're looking for cosine theta, but recall secand and cosine are related by being reciprocals to each other.
02:58
So we now know that cosine theta is four -fifths, taking the reciprocal of this line here, which puts four -fifths here.
03:09
Now we're ready to go minus from the line above, and we'll evaluate next, cosine of sine inverse of negative four -fifths.
03:19
Maybe let's do that evaluation here.
03:22
Cosine of sine inverse of negative four -fifths.
03:28
Again, our strategy is to let theta be equal to sine inverse of negative four -fifths, so that at the line above we're dealing with cosine of theta...