Find the acute angle solution to the following equation involving cofunctions. P is in degrees. sec ⁡ ( 4 P 5 + 20 ) = csc ⁡ ( 5 P 8 + 223 4 ) \sec\left(\frac{4P}{5}+20\right)=\csc\left(\frac{5P}{8}+\frac{223}{4}\right) sec ( 5 4 P + 20 ) = csc ( 8 5 P + 4 223 )

Welcome back, everyone. So let's give this problem a try. Here, we're asked to find the acute angle solution to the following equation involving cofunctions, and it's given to us that p is in degrees. So basically what this problem is asking us to do is to solve for this missing variable p. Now doing this is going to be a bit complicated because first off I noticed that we have some complicated fractions that we're dealing with within these trigonometric functions. But I also notice that we have 2 different trigonometric functions we're dealing with, a secant and a cosecant. So that means we're going to need to use cofunction identities to solve this problem. So let's see if we can go about doing this. Now the way I'm going to approach this equation is I'm going to start by taking the left side of the equation and using the cofunction identity, because I know that the cofunction of secant is cosecant. And when you're using a cofunction, you can take 90 degrees and subtract off the original angle that you have. So in this case, it would be 4p over 5 plus 20, because this is equivalent to getting the complementary angle of whatever you have within this trigonometric operation. And what we can do is set this equal to everything we have on the right side, so that will be the cosecant of 5p over 8, plus 223 over 4. Now from here, what we can do is we can simplify the left side of this equation. And I can take this minus sign, and I can distribute it into the parenthesis. So we're going to have the cosecant of 90 minus 4p over 5, minus 20, and this is going to equal the cosecant of 5pover8, plus 223 over 4. Notice the right side of this equation does not change. Now at this point, what I can do is combine this 90 and this negative 20, because 90 minus 20 is 70. So we're going to have the cosecant of 70 minus 4p over 5, and that's going to equal the cosecant of 5p over 8 plus 223 over 4. Now notice at this point we have the same trigonometric operation on both sides of the equation. So what that means is we can set the insides of the trig functions equal to each other. So what I can do is say that this 70 minus 4p over 5 is going to be equal to 5p over 8, plus 223 over 4. Now all I need to do is solve for this missing variable p using some algebra. And I'm gonna start by taking the 70 and subtracting it on both sides of the equation. That's going to give the seventies to cancel on the left side, leaving us with just negative 4p over 5. And that's going to equal 5p over 8, plus 223 over 4, minus 70. And minus 70, this can also be written as minus 70 times 4 over 4, because this will be the same thing as just 70, because these fours would naturally cancel. But what we can see here is that 70 times 4 would be 280, so we're going to have 280 over 4, and the reason that I'm doing this is because I'm trying to get these denominators equal to each other. Since we now have 4 in the denominators of both, we can just subtract the fractions that we have. So if I subtract the numerators, we'd have 223 minus 280, which comes out to negative 57. So rewriting this whole equation, we would have negative 4p over 5 equals 5p over 8, and this is gonna be minus 57, which would be the combination of the 223 minus 280 over 4. Now what I can do from here is take this 5p over 8, and subtract it on both sides of the equation. So I have minus 5p over 8 there, and minus 5p over 8 here. That is going to get the 5p over eights to cancel on the right side, and then we'll have 4p over 5 minuteus 5p over 8, and this is going to be what shows up on the left side. I'm gonna go ahead and write it over here so we have more space. So I have minus 4p over 5, minus 5p over 8, that's a 5 there, is equal to negative 57 over 4. Now to combine these fractions we need to get the same denominator. And I can do this by taking the top and bottom of this fraction on the left side, and multiplying it by 8. Because that's going to get us this denominator, and I could take this here, and I can multiply the top and bottom of this fraction by 5. So notice how we have an 8 times 5 and an 8 times 5, which is the same denominator. Now negative 4 times 8, well that's gonna give me negative 32. So we're going to have negative 32p over 5 times 8, which is 40, and then this is going to be minus, and then we have 5 times 5 which is 25p over 8 times 5 which is 40, that's all going to be equal to negative 57 over 4. Now from here, I can just combine these numerators since we have the same denominator in both now. So we have negative 32 minuteus 25, and negative 32 minus 25 is negative 57. So we're gonna have negative 57p over 40 is equal to negative 57 over 4. Now what I can do from here is cancel the negative signs, as we have 1 on both sides of the equation, and I can also take this 40 and move it to the other side by multiplying the left and right side of the equation by positive 40. It's gonna get these forties to cancel, leaving us with just 57p, and that's going to be equal to 57 times 40 over 4. Well, 4 divides into 40, 10 times, so we're going to have 57 times 10. And I'm not gonna combine these 2 just yet, because something that I noticed is since we have a 57 times 10 and a 57, we can just divide out that 57 on both sides. So we'll divide 57 on both sides of the equation, that'll do the 57s to cancel here, leaving us with just p. And then what I can do is cancel the 50 sevens there, leaving us with just 10 degrees. And 10 degrees is the answer to this problem. So that is how you can find angle p. Notice that the algebra was a bit tedious for this problem, but it becomes pretty straightforward to solve once you get the same trigonometric operation on both sides, and that's the main idea of these cofunction identities that we're using and solving these equations. So that's how you can solve this problem. Thanks for watching, and please let me know if you have any questions.