# How to determine a general arithmetic sequence formula for two intersecting trig function

I have equations out of two trigonometric functions.
For example

• $\cos(4\alpha$) = -$\sin(5\alpha)$
• $\tan(0.5\alpha$) = 2 $\sin(\alpha)$

How can I determine a general arithmetic sequence formula which gives me n-th positive angle where the two trig functions intersects?

By drawing the graph I found out the intersecting points..

1. 30, 70, 110... --> 30° + n * 40
2. 0, 120, 240... --> n * 120

Whereby n starts from 0.

Is there a better way to find these arithemetic sequence formula without drawing the graph?

$$\cos4\alpha=-\sin5\alpha=\cos\left(90^\circ+5\alpha\right)$$

$$\iff4\alpha=360^\circ m\pm\left(90^\circ+5\alpha\right)$$ where $m$ is any integer

Consider '+', '-' sign one by one.

For the second one use $$\sin2y=\frac{2\tan y}{1+\tan^2y}$$

• Seems complicated.. Could you please explain me how you've come up with this solution? Commented Aug 31, 2014 at 16:27
• @kk-dev11, Taking '-' sign, $$9\alpha=90^\circ(4m-1)\iff\alpha=10^\circ(4m-1)$$ etc. Commented Aug 31, 2014 at 16:29
• I see. Now I understand the final solution. What was the idea behind putting m into this cos4α=−sin5α=cos(90∘+5α) ? Commented Aug 31, 2014 at 16:35
• @kk-dev11, mathsfirst.massey.ac.nz/Trig/TrigGenSol.htm and $$\cos(90^\circ+y)=-\sin y?$$ Commented Aug 31, 2014 at 16:44
• The basic idea is that whenever $x$ is a solution of a trig equation, so are $x + 360^\circ$ and $x - 360^\circ;$ and therefore so are $x + 2\cdot360^\circ,$ $x - 2\cdot360^\circ,$ $x + 3\cdot360^\circ,$ and so forth. Commented Aug 31, 2014 at 16:55