Precalculus Trigonometry question

Question

An ant starts from the point (2, 0) and walks counterclockwise around the circle of radius 2 centimeters centered at the origin. The ant stops walking the first time it reaches a point whose y-coordinate is 1.5. How far does the ant walk?

Round your answer to three decimal places.

___cm

Answer 1

We can use inverse trig to find the measure of the central angle that subtends (cuts off) the arc the ant walks.

Even though degrees are more intuitive and the question can be done using degrees only, we will use radians for our units of angle measure as it makes the calculation of arc length super easy.

We place an angle with its vertex on the origin, with an initial ray passing thru the ant's starting coordinate, (2,0), and a terminal ray through the ant's stopping pt, (?,1.5). (Note we could easily calculate the x-coordinate but we don't need it so we don't care and won't bother.)

This central angle is also an acute angle in a right triangle with opposite leg length = 1.5 (y-coord of stop pt) and a hypotenuse length = 2 (the radius of the circle). So, this angle has a sine value = 1.5 / 2 = .75

Calculator time: sin^-1(.75) ~ .848 radians. The formula for arc length given a central angle in rads is s = rθ.

So the ant walks (2 cm)⋅(.848) ~ 1.696 cm

Answer 2

Draw and label a diagram!

The ant walks through an angle of θ

sin θ = 1/5 / 2

sin θ = 0.75θ

θ ≈ 0.8480^r

0.8480^r = 0.08480(2 cm)

The ant traveled 1.6961 cm