Trig Exam Question

https://www.quora.com/profile/Bravewarrior/p-158882520
Here is the question. I am stuck on part b.There are three solutions for x: 25,90 and 115. Can someone please explain how to get them? Thank you!

Sub part a) and factorise.

Oh wait I just realised cosx=0 then x=90 is the third solution 😬

We get (cos x)(cot x - cot (3x-50))=0 after the substitution....we can't eliminate any expression.
cos x= 0 or cot x - cot (3x-50)=0.
We can then use the identity for tan (a+b) to simplify the second expression.

Theres no need to use any identity for the second factor, just equate arguments of cot (with +/-k180).

could you please explain how? we have cot x and cot (3x-50)...are we going to rewrite these as one? if so, how?

You may be overthinking it
cot(x) = cot(3x-50)
so
x = 3x-50+k180
,,,
where k is an integer as cot is decreasing/invertible on 0..180 and repeats evey 180.

Oh right....I was overthinking it...thnx

tbh its one that crops up every now and then in various disguises, so solving
cos(x) = sin(2x)
(or similar) can be done using cos(x) = sin(90-x) and then arcsin both sides. Its a useful "trick" to watch out for

Its best to just post hints, rather than solutions, as per the sticky at the top of the forum. One small thing though would be that cos(x)=0 has solutions every 180 as its simply the axis crossing points so 90,-90,270, .... The other solutions obviously lie outside the domain for this question though.

Thanks to point that out. Did not notice the cosine glitch. Since this site doesn't directly support image and does not have a preview feature, so it still just show 2 links after spending some time figuring how to do it. As for the solution vs hints, my personal view is that fully understand the background is better than just get the answer right. Well this site is generally designed for hints so I think I am in the wrong place. Good luck to all students' endeavors!

Thank you so much! 😃

You are welcome. This site is not really user friendly especially for high quality posts with comprehensive background explanations. For students troubled by trigonometry or any other math/physics areas, I would suggest to start with the origin. Once you establish fundamentals, you'll be able to derive everything else instead of memorizing the rules. That's why so many students rely on other sources like YouTube channels to deepen their understanding, as many professionals will explain the principles using different ways than in the school. Good luck with your exam and get all A*! 😁