Advanced Mathematics and Trigonometry Study Guide

  • Law of Cosines for Side Lengths:

    • Problem 8c: Given a triangle with sides of 88 and 66 and an included angle of 45°45^\text{°}.

      • Calculation: v2=82+62−2(8)(6)cos(45°)v^2 = 8^2 + 6^2 - 2(8)(6)\text{cos}(45^\text{°}).

      • v2=64+36−96×sqrt(2)21v^2 = 64 + 36 - 96 \times \frac{\frac{\text{sqrt}(2)}{2}}{1}.

      • v2=100−48sqrt(2)v^2 = 100 - 48\text{sqrt}(2).

      • v evaluation pendingv \text{ evaluation pending}.

    • Problem 8d: Given a triangle with sides of 77 and 1010 and an included angle of 30°30^\text{°}.

      • Calculation: v2=72+102−2(7)(10)cos(30°)v^2 = 7^2 + 10^2 - 2(7)(10)\text{cos}(30^\text{°}).

      • v2=49+100−140×sqrt(3)2v^2 = 49 + 100 - 140 \times \frac{\text{sqrt}(3)}{2}.

      • v2=149−70sqrt(3)v^2 = 149 - 70\text{sqrt}(3).

      • v evaluation pendingv \text{ evaluation pending}.

  • Area of Triangle RST:

    • Problem 9c: Triangle RST with sides 99 and 1212 and an angle of 30°30^\text{°}.

      • Calculation: Area=12(9)(12)sin(30°)\text{Area} = \frac{1}{2}(9)(12)\text{sin}(30^\text{°}).

      • Area=12(108)(0.5)=27\text{Area} = \frac{1}{2}(108)(0.5) = 27.

    • Problem 9d: Triangle RST with sides 44 and 1010 and an angle of 60°60^\text{°}.

      • Calculation: Area=12(4)(10)sin(60°)\text{Area} = \frac{1}{2}(4)(10)\text{sin}(60^\text{°}).

      • Area=20×sqrt(3)2=10sqrt(3) (approximately 17.32)\text{Area} = 20 \times \frac{\text{sqrt}(3)}{2} = 10\text{sqrt}(3) \text{ (approximately } 17.32\text{)}.