Exercises: Solve Trigonometric Equations

The function $\arcsin$ has a restricted domain. What is the range of $\arcsin$?

A model gives $h(t) = 12\sin(\pi t / 6) + 15$. Set $h(t) = 21$. After isolating the sine, what value does $\sin(\pi t/6)$ equal?

Enter as a fraction (e.g., 1/2).

A student evaluates $\arcsin(0.6)$ on a calculator and gets $36.87$. What is the likely error?

A tide model is $H(t) = 3\cos(\pi t/6) + 5$, where $t$ is hours. Set $H = 7$.

After isolating the cosine, what value does $\cos(\pi t/6)$ equal?

Enter as a fraction (e.g., 2/3).

A Ferris wheel model gives $h(t) = 20\sin(\pi t/4) + 25$. You want to find when $h = 40$.

After isolating the sine expression: $\sin(\pi t/4) = ?$

Enter as a decimal rounded to two decimal places (e.g., 0.75).

Given $\sin(\theta) = 0.75$ (from the Ferris wheel problem above), find the principal value $\theta = \arcsin(0.75)$.

Use a calculator (radian mode). Round to four decimal places.

You found $\sin(\pi t/4) = 0.75$, and the principal solution gives $\pi t/4 \approx 0.8481$.

Within one period of length 8 (since the period is $2\pi / (\pi/4) = 8$), there is a second value of $\pi t/4$ that also gives $\sin = 0.75$. What is it?

From the Ferris wheel problem: $\pi t/4 = 0.8481$ gives $t_1$, and $\pi t/4 = 2.2935$ gives $t_2$.

What is $t_1$ (the first time the Ferris wheel reaches height 40)? Round to two decimal places.