# Trigonometric eliminations

These are a few problems which I wasn't able to do. I am new to these trigonometric eliminations.

I don't really know how to start these problems. I couldn't get pass the first step in some of them..Any help will be appreciated.

(1). Eliminate $\theta$ from the following

$\sin\theta - \cos\theta = p\quad$ and $\ cosec\theta - \sin\theta = q$

I tried adding $p$ and $q$ and squaring them but couldn't do it further.

(2). Eliminate $\theta$ from the following

$a\cos2\theta = b\sin\theta$, and $c\sin2\theta = d\cos\theta$

In this I tried :

$2c\sin\theta\ cos\theta = d\cos\theta$

$d = 2c\sin\theta$

Now

$a\cos2\theta = b\sin\theta \ \div d = 2c\sin\theta$

gives $\frac {a\cos2\theta}{d} = \frac b{2c}$

I dont know how to solve further than this. Please help me with these 2 questions.

## 1 Answer

For $(2),$ as $d=2c\sin\theta\implies\sin\theta=?$

and $b\sin\theta=a\cos2\theta\implies b\sin\theta=a(1-2\sin^2\theta)$

Set the value of $\sin\theta$

For $(1),q=\csc\theta-\sin\theta=\dfrac{\cos^2\theta}{\sin\theta}\implies q\sin\theta=\cos^2\theta=1-\sin^2\theta$

$\sin^2\theta+q\sin\theta-1=0\ \ \ \ (i)$

and $\sin\theta-\cos\theta=p\iff\cos\theta=p-\sin\theta$

Square to form a Quadratic Equation$(ii)$ in $\sin\theta$

Solve $(i),(ii)$ for $\sin\theta,\sin^2\theta$

and finally use the identity $(\sin\theta)^2=\sin^2\theta$

• Thank you sir! Any idea for (1) ? – Shubham Sep 29 '14 at 16:56
• @Shubham, Added $(1)$ – lab bhattacharjee Sep 29 '14 at 17:00
• By solving (i),(ii) what do you mean? Do I find the values of $sin\theta$ for both equations separately using quadratic formula and then divide them or do I add subtract (i) and (ii) to get rid of $sin\theta$ ? The former is lengthy but worked but I couldn't get rid of $sin\theta$ in the latter. – Shubham Sep 29 '14 at 17:16
• @Shubham, Do you know how to solve simultaneous equations? – lab bhattacharjee Sep 29 '14 at 18:12
• Can you be more specific? Like equations in 2 variables, parabola intersections etc? – Shubham Sep 29 '14 at 18:16