# Chapter 3 - Radian Measure and the Unit Circle - Section 3.3 The Unit Circle and Circular Functions - 3.3 Exercises - Page 125: 90

The shortest day in Minneapolis is $~~8.6~hours$ The longest day in Minneapolis is $~~15.4~hours$

#### Work Step by Step

We can convert $44.88^{\circ}$ to units of radians: $L = (44.88^{\circ})(\frac{\pi~rad}{180^{\circ}}) = 0.7833~rad$ We can convert $-23.44^{\circ}$ to units of radians: $D = (-23.44^{\circ})(\frac{\pi~rad}{180^{\circ}}) = -0.4091~rad$ We can calculate the shortest day in Minneapolis: $cos(0.1309~H) = -tan~D~tan~L$ $cos(0.1309~H) = -tan~(-0.4091)~tan~(0.7833)$ $cos(0.1309~H) = 0.431746$ $0.1309~H = cos^{-1}(0.431746)$ $0.1309~H = 1.12436$ $H = \frac{1.12436}{0.1309}$ $H = 8.6~hours$ The shortest day in Minneapolis is $~~8.6~hours$ We can calculate the longest day in Minneapolis: $cos(0.1309~H) = -tan~D~tan~L$ $cos(0.1309~H) = -tan~(0.4091)~tan~(0.7833)$ $cos(0.1309~H) = -0.431746$ $0.1309~H = cos^{-1}(-0.431746)$ $0.1309~H = 2.017224$ $H = \frac{2.017224}{0.1309}$ $H = 15.4~hours$ The longest day in Minneapolis is $~~15.4~hours$

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