# Thread: doing things in the right order

1. ## doing things in the right order

Hi folks,

I am trying to find the general solution to a trig equation:

I get to $5 sin (\theta + 53.1\circ) = 1$

so $(\theta + 53.1) = sin^{-1}(\frac{1}{5}) = 11.5\circ$

so I think the general solution is $\theta + 53.1 = 180\circ n + (-1)^n 11.5\circ$

i.e. $\theta = 180\circ n + (-1)^n -41.6\circ$

the answer in my maths books is $360\circ n + 115.3\circ , 360\circ n - 41.6\circ$

I can understand the second solution but not the first.

I could have simplified the second equation to $\theta = -41.6$ with solutions at 41.6 and 221.6

Or I could have $\theta + 53.1 = 11.54$ and $(180 - 11.5)$

I am not sure which order to do things.

2. ## Re: doing things in the right order

$\sin(t+53.1)=\dfrac{1}{5} \implies (t+53.1)$ is an angle in quadrant I or II.

$t+53.1 = \arcsin\left(\dfrac{1}{5}\right) + k \cdot 360$ where $k \in \mathbb{Z}$

$t = -41.56 + k \cdot 360$

$t+53.1 = 180 - \arcsin\left(\dfrac{1}{5}\right) + k \cdot 360$

$t = 115.36 + k \cdot 360$

3. ## Re: doing things in the right order

Originally Posted by s_ingram
Hi folks,

I am trying to find the general solution to a trig equation:

I get to $5 sin (\theta + 53.1\circ) = 1$

so $(\theta + 53.1) = sin^{-1}(\frac{1}{5}) = 11.5\circ$

so I think the general solution is $\theta + 53.1 = 180\circ n + (-1)^n 11.5\circ$
Here is your error: (-1)^n(11.5)- 53.1 is not equal to (-1)^n(11..5- 53.1). The "(-1)^n" does not multiply -53.1.

i.e. $\theta = 180\circ n + (-1)^n -41.6\circ$

the answer in my maths books is $360\circ n + 115.3\circ , 360\circ n - 41.6\circ$

I can understand the second solution but not the first.

I could have simplified the second equation to $\theta = -41.6$ with solutions at 41.6 and 221.6

Or I could have $\theta + 53.1 = 11.54$ and $(180 - 11.5)$

I am not sure which order to do things.