Question

Given point (-8,8) how do you find the distance of the point from the origin, then find the measure of the angle in standard position whose terminal side contains the point?

Answer 1

Measure of the angle `theta = 135^@` or `((3pi)/4)^c`

Explanation:

Distance formula : `d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

Given point `A (-8, 8)` and Origin `O (0,0)`

`vec(OA) = r = sqrt ((-8)^2 + 8^2) = sqrt 128 = 8sqrt2`

`x = r cos theta`, `y = r sin theta`

`- 8 = 8 sqrt2 cos theta`

`cos theta = -cancel8 / (cancel8 sqrt2) = -1/sqrt2` or `theta =color(purple)( - pi/4 = (pi - pi/4) = (3pi)/4`

Similarly, `8 sqrt2 sin theta = 8`

`sin theta = 8 / (8 sqrt2) = 1/sqrt2` or `theta = pi/4` or = `pi - pi/4) = (3pi)/4`

Slope of the line `m = tan theta = -8 / 8 = -1`

`x` `(-)` and `y` `(+)` in II quadrant.

Hence `theta = tan ^-1 (-8/8) = tan^-1 -1 = -1 = (3pi)/4`

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