I am having issues figuring out the correct steps to solve this and a walk through would be appreciated... Please help!

Question

A metal plate has the form of a quarter circle with a radius of R = 100 cm. Two 3 cm holes are to be drilled in the plate r = 95 cm from the corner at  and  as shown above. To use a computer controlled milling machine you must know the Cartesian coordinates of the holes. Assuming the origin is at the corner what are the coordinates of the holes (x₁, y₁) and (x₂, y₂)? Round your answer to 3 decimal places

How would I go about solving this (steps please) ?

Answer 1

A problem (intentionally?) written to seem more difficult than it is.

Forget everything about the quarter circle, radius of 10, 3 cm holes.

As stated the corner is at the origin.

Draw two lines both 5 units - one at 30° the other at 60°.

Drop perpendulars to the x-axis.

You now have not one but two 30-60-90 triangles with hypotenuses of 5.

Calculate the lengths of the legs an you have the coordinates.

This is more of a Geometry problem than Pre-Calc or Trig.

Answer 2

Without the figure I cannot answer your specific question,

but let me tell you the steps in general.

Suppose you are to find Cartesian coordinates (x, y) of a point P,

and that point P is described in so called "polar coordinates".

That means you are given

1) the length c of the line segment connecting the point P with the origin, and

2) the angle α between that line segment and the x-axis.

In your example, you are to find Cartesian coordinates of two points,

one has α = 30° and the other α = 60°.

Both have the same distance c from the origin, which is probably close to 95 cm,

but I cannot tell without the figure.

To find the Cartesian coordinates the general point P, drop a line from P perpendicular

to the x-axis.

That defines a right angle triangle, whose two sides are the Cartesian coordinates x and y, and whose hypotenuse is c.

By definition of sine and cosine

sin(α) = y/c

cos(α) = x/c

From that

y = c×sin(α)

x = c×cos(α)

Therefore the Cartesian coordinates are (c×cos(α), c×sin(α)).