Sinusoidal Function word problems: month and temperature $T(t) = 14.9 \sin {\pi \over 6}(t-3) +13$

Question

The averge monthly maximum temperature of New York can be modelled by $T(t) = 14.9 \sin {\pi \over 6}(t-3) +13$ where $T$ is the temperature in Celsius and $t=0$ represents January. Predict when the temperature is $0$ $^{\circ}$C

I tried this:

$$0 = 14.9 \sin {\pi \over 6}(t-3) +13\\ -13 = 14.9 \sin {\pi \over 6}(t-3)\\ -27.9 = \sin {\pi \over 6}(t-3)$$ Let ${\pi \over 6}(t-3)= \theta$

$$\sin \theta = -27.9$$ $$*error$$

Then I was going to substitute the value of $\theta$ back into ${\pi \over 6}(t-3)= \theta$ to obtain the $t$ value.

How do I do this question?

Answer 1

It's just a simple arithmetic error. You should divide instead of subtract

$$ \sin \left(\frac{\pi}{6}(t-3)\right) = \frac{13}{14.9} = 0.872 $$

And then use inverse sine. You should know something is wrong since $-1 \le \sin\theta \le 1$