# Math Help - Trigonometric equations?

1. ## Trigonometric equations?

The number of solutions of tan(5π cos θ) = cot (5π sin θ) in (0,2π) is?
A) 28
B) 14
C) 4
D) 2

2. Originally Posted by fardeen_gen
The number of solutions of tan(5π cos θ) = cot (5π sin θ) in (0,2π) is?
A) 28
B) 14
C) 4
D) 2
Personally, I would graph it. And then you can count the number of times $f(x)=\tan(5\pi\cos(\theta))-\cot(5\pi\sin(\theta))$

3. No analytical method to solve this? In the case i go in for a graph, can u suggest the name of any good graphing calculator online?

Thanks.

4. Hello
Originally Posted by fardeen_gen
The number of solutions of tan(5π cos θ) = cot (5π sin θ) in (0,2π) is?
This equality can be written

$\frac{\sin(5\pi\cos\theta)}{\cos(5\pi\cos\theta)}-\frac{\cos(5\pi\sin\theta)}{\sin(5\pi\sin\theta)}= 0$

If $\sin(5\pi\sin\theta)\neq 0$ and $\cos(5\pi\cos\theta)\neq 0$, this equation is the same as $\sin(5\pi\cos\theta)\sin(5\pi\sin\theta)- \cos(5\pi\sin\theta)\cos(5\pi\cos\theta)=0
$

Since $\cos(a+b)=\cos a \cos b-\sin a\sin b$ the equation becomes $\cos\left[5\pi(\sin \theta+\cos \theta) \right]=0$.

EDIT : To find the number of solutions you may need $\sin \theta + \cos \theta = \sqrt{2}\cos\left(\theta-\frac{\pi}{4}\right)$.