Find the first 4 times when a robotic arm is at h=0

Question

Hi I need help with this simple question that I just can't seem to wrap my head around.

The arm of a robot moves up and down such that the height h at time t is given by h=sin^2 t - 0.2 sin t. Find the first 4 times when the arm is at h = 0.

The answer is t=0, 0.201, 2.94, and pi. But how do we arrive to these 4 numbers?

Answer 1

Try the substitution

x = sin(t)

Then the equation becomes

h = x² - 0.2x = x(x - 0.2)

Set h=0 and solve for x. Right away, you can see x=0 and x=0.2 are solutions.

x = 0 ⇒ sin(t) = 0 ⇒ t = 0, π, 2π, 3π, ...

x = 0.2 ⇒ t = arcsin(0.2) = 0.201 (with roundoff)

Keep in mind the identity

sin(y) = sin(π - y),

so π-0.201 must also be a solution.

π - 0.201 = 2.94 (with roundoff)

The first 4 solutions for t are 0, 0.201, 2.94, and π.

By the way, thank you for posting an interesting problem like this.