Question

The angle θ=π3 is in standard position and has a terminal side that coincides with the graph of a proportional relationship represented by y=kx.

a. Find the constant of proportionality, k.

b. Write an ordered pair that lies on the graph of y=kx.

Answer 1

Part a

theta = pi/3

sin(pi/3) = sqrt(3)/2

cos(pi/3) = 1/3

tan(angle) = opposite/adjacent = rise/run = slope

tan(pi/3) = sin(pi/3)/cos(pi/3) = (sqrt(3)/2)/(1/2) = sqrt(3)

The value of k is k = sqrt(3)

This is also the slope the line y = kx because it is in the form y = mx+b with m = sqrt(3) and b = 0.

===============================================

Part b

Pick any value you want for x, and plug it in to find the paired value of y

if x = 0, then y = 0 because

y = sqrt(3)*x = sqrt(3)*0 = 0

So (0,0) is one ordered pair on the line

---------

If x = 1, then y = sqrt(3), since,

y = sqrt(3)*x = sqrt(3)*1 = sqrt(3)

We can say ( 1, sqrt(3) ) is another ordered pair

---------

If x = sqrt(3), then,

y = sqrt(3)*x = sqrt(3)*sqrt(3) = sqrt(3*3) = sqrt(9) = 3

Therefore, ( sqrt(3), 3 ) is another ordered pair

There are infinitely many ordered pairs on this line.