Thus, that hypotenuse has length 1000 feet. This leads to the
very simple equation x = 1000 sin(60°). In either case, you will
get x
≈
866.03 feet and so h
≈
871.03 feet.
[Warning: The cute
isosceles triangle thingy doesn’t usually happen. The other
approach is more general.]
14. (5 pts.)
If the polar coordinates of a point are given by
(r,
θ
) = (9.5,110°), find the rectangular coordinates for the
point. In doing this, make clear which values are exact and
which are approximations.
(x,y) = ((9.5)cos(110°),(9.5)sin(110°))
≈
(3.25, 8.93)
15. (5 pts.)
If the rectangular coordinates of a point are
given by (x,y) = (5,5
√
3), obtain polar coordinates for the
point.
It’s easy to see r
2
= 100. Thus, use r = 10, to keep things
simple. It turns out that the reference angle
θ
r
satisfies the
equation tan(
θ
r
) = 3
1/2
. Thus, because the point lies in the
third quadrant, we may use either
θ
= 240° or
θ
= 4
π
/3. From
here, it is easy to list all pairs (r,
θ
) that represent the
point.
16. (10 pts.) (a) Obtain all solutions to the equation below,
and then (b) list the solutions
θ
with 0
≤
θ
< 2
π
.
2 sin
2
(
θ
) + 3 sin(
θ
) + 1 = 0
(a) The given equation is equivalent to
(2sin(
θ
) + 1)(sin(
θ
) + 1) = 0,
which is equivalent to sin(
θ
) = 1/2 or sin(
θ
) = 1. All
solutions to sin(
θ
) = 1/2 are given by
θ
= 7
π
/6 + 2k
π
or
θ
= 11
π
/6 + 2k
π
, k any integer, and all solutions to sin(
θ
) = 1
are given by
θ
= 3
π
/2 + 2k
π
, k any integer. (b) The solutions in
the desired interval are 7
π
/6, 3
π
/2, and 11
π
/6.
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 Spring '08
 Storfer
 Pythagorean Theorem, Sin, pts, triangle

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