e system of equations
tan(60°) = x/d and
tan(30°) = x/(d + 1000).
Solving for x
yields x = 1000(tan(30°))/(1 - cot(60°)tan(30°)).
you could recognize that, here, the hypotenuse of the triangle
created by the 60° sighting is one of the two equal sides of the
isosceles 30°,30°,120° triangle formed from the 30° sighting.
Thus, that hypotenuse has length 1000 feet. This leads to the
very simple equation x = 1000 sin(60°).
In either case, you will
866.03 feet and so h
[Warning: The cute
isosceles triangle thingy doesn’t usually happen.
approach is more general.]
14. (5 pts.)
If the polar coordinates of a point are given by
) = (9.5,110°), find the rectangular coordinates for the
In doing this, make clear which values are exact and
which are approximations.
(x,y) = ((9.5)cos(110°),(9.5)sin(110°))
15. (5 pts.)
If the rectangular coordinates of a point are
given by (x,y) = (-5,-5
3), obtain polar coordinates for the
It’s easy to see r
Thus, use r = 10, to keep things
It turns out that the reference angle
Thus, because the point lies in the
third quadrant, we may use either
= 240° or
here, it is easy to list all pairs (r,
) that represent the
16. (10 pts.)
(a) Obtain all solutions to the equation below,
and then (b) list the solutions
) + 3 sin(
The given equation is equivalent to
) + 1)(sin(
which is equivalent to sin(
) = -1/2 or sin(
) = -1.
solutions to sin(
) = -1/2 are given by
, k any integer, and all solutions to sin(
are given by