How do you solve for t in 50 = 90sin(2pi*t / 26 + 3pi / 2) +115

Question

50 = 90sin(2pi*t / 26 + 3pi / 2) +115

I am not allowed to solve this by drawing a graph. Hence, I have to show working.

I understand that I have to carry over the 115 and divide both sides by 90 which will get me

sin(2pi*t/26 + 3pi/2) = -13/18

For the next step, I think I am wrong as I did this

2pi*t/26 + 3pi/2 = 0.807

26(0.807 - 3pi/2) / 2pi

=-16.16

-16.16 + pi = -13

2pi - (-16.16) = 22.44

This is clearly wrong as the answers are 3.16 and 22.8 according to desmos. Is there a concept I am unaware of? Regardless, it would be best if you could show me the correct working anyway.

Answer 1

Nice job of getting .807, but you forgot the - sign. You will then get -22.8 You left off the - sign on the desmos value.

Actually, a more precise value (so you can plug in and check) is -22.8394296944176

The solution would actually be =13/PI()*SIN^-1(-13/18)-19.5 (or you can write -13/pi()*SIN^-1(13/18)-19.5; I prefer the former; Excel uses asin for arcsin)

As we see sin(2 pi t/26), the period is 26. 26 + -22.8394296944176 = 3.1605703055824

I thus match the other desmos value.