Online Course Discussion Forum
MC II B: I am not sure how to start questions 4.25, 4.26, and 4.30...
Could you possibly provide some minor hints to start the problems? Thank you.
For 25, try factoring (remember $\cos^n(x)$ means $(\cos(x))^n$) and see if you then spot a way to use trigonometric identities you've learned so far (like $\sin^2(x) + \cos^2(x) = 1$, $\sin(2x) = 2\sin(x)\cos(x)$, etc.).
For 26, recall that supplementary angles have the same $\sin$, so of $\alpha + \beta = 180^\circ$, then $\sin(\alpha) = \sin(\beta)$.
For 30, try using the result from example 4.10: in a triangle $ABC$ with sides $a,b,c$ we have $$\dfrac{a-b}{a+b} = \tan\left(\dfrac{A-B}{2}\right)\tan\left(\dfrac{C}{2}\right).$$
Let us know if these helped, or if you need additional hints.
Unfortunately, I am still stuck and may require additional hints.
Can you tell us what have you tried so far?
Yes. I'm stuck at the trigonometric identities because I can't seem to find any that satisfy and I have tried to add the two arcsines of the sines provided but it is not seeming to work out cleanly. Finally, I'm not sure how the third hint applies to the problem.
For 25, after factoring, try using the double angle trigonometric identities.
For 30, since angles do not change when we have similar triangles, it is safe to assume $c=1$, and so $b = 2+\sqrt{3}$. Using the previous hint you can calculate $\tan\left(\dfrac{B-C}{2}\right)$, and then use that value of $\tan$ to find what $B-C$ is.
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