How to Find Unknown Angles

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    • 1). Know and understand what numbers you have, what they mean and what the formulas are for solving the angles. Let's say that you don't know any of the interior angles; all you know are the exterior sides: Side a = 3, Side b = 4 and Side c = 6.

    • 2). Understand that you always need to find the largest angle first (because only one angle can be greater than 90 degrees). In this example, Side c is the largest. Use the cosine rule to find angle C: c^2 = b^2 + a^2 - 2ba(cosC)

      Add 2ba(cosC) from both sides to pull it to the left of the equation and subtract c^2 from both sides to pull it to the right. Also, reverse b^2 with a^2 so that a^2 is in the front.

      2ba(cosC) = a^2 + b^2 - c^2

      Now divide 2ba from both sides.

      cosC = a^2 + b^2 - c^2 / 2ab

      Plug in your numbers and solve.

      cosC = 3^2 + 4^2 - 6^2 / 2 x 3 x 4
      = (3 x 3) + (4 x 4) - (6 x 6) / 2 x 3 x 4
      = 9 + 16 - 36 / 24
      = -11 / 24

      cosC = - 0.46

      Now find the inverse cos of -0.46 by using your calculator.

      C = cos^-1 (-0.46)

      Angle C = 117.279 degrees

    • 3). Find angle A using the sides of the triangle and angle C. To find angle A, you will use the sine rule: a / sin A = c / sinC

      Multiply a from both sides to solve for sinA. Also, reverse c with sinC.

      sinA = a sinC / c

      Plug in your numbers to solve.

      sin A = 3 x sin 117.279 / 6

      Input 117.279 then "sin" on your calculator.

      sin A = 3 x .889 / 6
      sin A = 2.667 / 6
      sin A = 0.445

      Determine the inverse sin of 0.445 using your calculator.

      A = sin^-1 (0.445)

      Angle A = 26.423 degrees

    • 4). Find angle B by using the Sum of Internal Angles rule. All the internal angles have to add up to 180. The actual equation is: A + B + C = 180.

      Subtract (A + C) from both sides to solve for B.

      B = 180 - (A + C)
      B = 180 - (26.423 + 117.279)
      B = 180 - 143.702
      B = 36.298

      Angle B = 36.298 degrees

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