If $\alpha$ and $\beta$ are distinct roots of the equation $p \cos(\theta) + q \sin(\theta) = r,$ then show that $\tan ((\alpha+\beta)/2) = q/p.$

Question

I tried dividing both sides by $\cos$ and then squaring them. Then I converted $\sec^2(x)$ to $\tan^2(x)$ but still no luck even after substituting values.

https://math.stackexchange.com/questions/1833153/if-sin-theta-sin-phi-a-and-cos-theta-cos-phi-b-then-sin-thet and https://math.stackexchange.com/questions/452631/if-sin-theta-sin-phi-a-and-cos-theta-cos-phi-b-then-find-tan-frac — lab bhattacharjee, Apr 26 '18 at 03:53
See my solution to the question "Cosine of the sum of two solutions of trigonometric equation $a\cos\theta+b\sin\theta=c$". — Blue, Apr 26 '18 at 04:04

score 0 · Answer 1 · answered Apr 25 '18 at 17:23

0

hint...Put $t=\tan\frac 12\theta$ so that $\cos\theta=\frac{1-t^2}{1+t^2}$ and $\sin\theta=\frac{2t}{1+t^2}$.

Then consider the sum and product of the roots of the quadratic in $t$

answered Apr 25 '18 at 17:23

David Quinn

35,087

score 0 · Answer 2 · answered Apr 26 '18 at 03:57

0

$$p(\cos\beta-\cos\alpha)=q(\sin\alpha-\sin\beta)$$

Now apply this

answered Apr 26 '18 at 03:57

lab bhattacharjee

279,016

If $\alpha$ and $\beta$ are distinct roots of the equation $p \cos(\theta) + q \sin(\theta) = r,$ then show that $\tan ((\alpha+\beta)/2) = q/p.$

2 Answers2

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