# Solving $52\sin\left(\frac{2\pi}{365}t\right)+728=750$

I have seen the same formula posted, but the questions are all slightly different from what I am looking for.

Here is the equation I am trying to solve from a Khan Academy Exercise.

$$52\sin\left(\frac{2\pi}{365}t\right)+728=750$$

I can simplify the equation to the following but then get the incorrect result?

$$t = \frac{365}{2\pi}(0.4368+2\pi n)$$

$$250 + 365n \quad\text{(incorrect)}$$

$$25 + 365n \quad\text{(correct)}$$

I have been stuck trying to figure out what I am doing wrong here. If anyone has any ideas please let me know!

• $t=\frac{365}{2\pi}(0.4368+2\pi n)\approx 25 + 365n$ , I don't know how can you get the result $250+365n$
– Hugo
Commented Mar 25, 2021 at 5:11
• When entering it into my calculator without brackets, I was incorrectly using brackets which messed up the order of operations. Commented Mar 27, 2021 at 1:16

It seems that

$$\frac{750-728}{52} = \frac{22}{52} = 0.423$$

So the solution is

$$\frac{2\pi}{365}t +2n\pi = 0.4367$$

Now you can rewrite this as

$$t+365n = \frac{0.4367}{2\pi}\cdot 365 = 25.37$$

$$t = 25.37 -365n$$