# Inverse of a function that includes sin

I'm learning calculuus and currently struggling with Mooculus text book.

I am unable to solve this problem, basically because I never understood trigonometric functions in high school. Chapter 0.2 Inverses of Functions, exercise # 5 states:

The height in meters of a person off the ground as they ride a Ferris Wheel can be modeled by

$h(t) = 18 · \sin (\frac{\pi · t}{7} ) + 20$

where t is time elapsed in seconds. If h is restricted to the domain [3.5, 10.5], find and interpret the meaning of $h^{-1}(20)$

Answer says:

$h^{-1}(20) = 7$ This means that a height of 20 meters is achieved at 7 seconds in the restricted interval. In fact, it turns out that $h^{−1}(t) = 7 · (\frac{\pi − arcsin(\frac{(t − 20)}{18})}{\pi})$ when h is restricted to the given interval.

Question 1) How did two $\pi$ appeared into the answer? I've been reading about trigonometric functions on internet but they all talk about hypotenuse, adjacent and opposite sites. I know the definition of sin and arcsin as its inverse function. But I'm still unable to understand how was calculated inverse of the function.

Question 2) How are trigonometric functions related to circles (Ferris wheel)?

Question 3) How is given domain used to calculate answer?

I would appreciate if someone tells what do I need to learn / know / understand in order to resolve this problem so I can read it on my own.

Thanks

• Son/cos are defined by using the unit CIRCLE – Zelos Malum Aug 28 '16 at 6:12
• You should understand that two angles whose difference is $2\pi$ are geometrically congruent but still they are different. So when you know $\sin t = x$ you are identifying a class of angles: $t = \sin^{-1} x + 2k\pi$ where $k$ can be any integer. When a problem gives you a range you can find what particolar value to give to $k$. – N74 Aug 28 '16 at 7:30