# Thus that hypotenuse has length 1000 feet this leads

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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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