00:01
For this problem, we're given that the sign of alpha is 5 .13s, where alpha is between 0 and pi over 2, and beta, cosine of beta is 2 squared of 85 over 85, where beta is between negative pi over 2 and 0, or a fourth quadrant angle.
00:16
And we're asked to find these values right here.
00:24
So in order to be able to define the sign of alpha plus beta or the cosine of alpha plus beta and so on, we need to find out with the cosine of alpha and the tangent of alpha.
00:32
And we need to find the sign of alpha, beta and the tangent of beta.
00:37
So we'll use these equations right here in order to help us do that.
00:41
So let's begin with the first one, sine squared of x plus cosine squared of x equals one.
00:45
So we do know what the sign of alpha is.
00:49
So sine squared of alpha, 513 squared.
00:55
That's what sign squared means.
00:57
It's square the value of the sign plus cosine squared of alpha equals one.
01:06
If i squared, 513th, i square the 5, 25, i square the 13169 plus cosine squared of alpha equals.
01:17
Let me write this 1 is 169 over 169.
01:23
And now if we subtract 25, 169th from both sides, we get that the cosine squared of alpha equals 144 over 169.
01:38
And then if we apply a square root to get rid of the squared there, we do a two.
01:44
Remember to consider both pause for negative.
01:47
So cosine of alpha is equal to.
01:51
And we come back here and we remember that alpha is a first quadrant angle.
01:54
So the cosine of a first quadrant angle is going to be positive.
01:57
So we'll just keep the positive one, 12 over 13.
02:03
Now let's figure out what this tangent value is here.
02:05
We see that the tangent is the sign over the cosine.
02:10
So tangent of alpha is going to be sign of alpha, which is 5 .13s over the cosine of alpha, which is 1213s.
02:23
And the 13th cancel out here and we get that the tangent of alpha is equal to 512s.
02:33
All right, so we have cosine of alpha and tangent of alpha now.
02:36
So let's go ahead and calculate those values, the other two values for beta.
02:45
So again, using this top equation here of sine squared x plus cosine squared x equals 1, we are calculating for sine of beta, sine squared of beta, plus i'm going to square the value of cosine, so 2 squared of 85 over 85 squared equals 1.
03:15
So that's going to give me sine squared beta equals 340, sorry, plus sine squared beta plus 340 over 85 squared is 72225 equals.
03:37
And again, i'll write this 1 is 72225 over 72225.
03:44
Subtracting 340 over 7 ,025 from both sides i get that the sine of beta, sine squared beta is equal to 6885 over 7225 and again applying a square root and again don't forget to consider both positive so we're going to get here that the sign of beta equals and scroll back up here and we see that beta is in the fourth quadrant and only cosine and secant are positive in the fourth quadrant.
04:34
So the sign of beta will be negative.
04:37
And simplifying that square root, we get nine times the square root of 85 over 85.
04:48
And then now calculating the tangent.
04:52
So the tangent of beta is going to be equal to sign.
04:59
So negative 9, square of 85 over 85.
05:05
Over the cosine, which is two, square root of 85 over 85.
05:14
Well, both denominators have an 85, and both numerators have a square root of 85.
05:19
So that's simple.
05:19
All those simplified just to negative nine halves.
05:26
All right, now we have all these values that we need to calculate.
05:29
So let me scroll down here a little bit further.
05:33
And we can see here, these are what we need to calculate.
05:43
So when we find the sign of alpha plus beta, we'll use this formula right here.
05:51
So the sign of alpha plus beta is going to be, we just have to fill in the sign of alpha, the cosine of beta, the cosine of alpha, and the sine of beta.
05:59
So let's scroll just for a second here.
06:08
Sign of alpha is 5 .13s.
06:14
So sign of alpha is 513ths.
06:20
Cosine of alpha is 12 thirteenth tangent of alpha is 5 twelfths and cosine of beta was two squared of 85 over 85 sine of beta was negative 9 squared of 85 over 85 and then the tangent of beta is equal to negative 9 has...