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How to Calculate Angles Based on Sides

The Pythagorean theorem tells you that for any right triangle, the square of the longest side of the triangle, which is its hypotenuse, equals the sum of the squares of the two sides. This is often written c^2 = a^2 + b^2. The law of cosines looks almost the same, with one term added to the right side of the expression: c^2 = a^2 + b^2 - 2abCosC. Use the law of cosines and a similar law of sines to find the angles of triangles when you only know the measurements of the sides of the triangle.

Things You'll Need

Ruler
Scientific calculator

Instructions

- 1
  Place the triangle you want to analyze in front of you so that one side of the triangle is parallel to the table edge closest to you. Label the side of the triangle closest to you "a." Label the next side, moving in a counterclockwise direction, "b" and label the remaining side "c."
- 2
  Label the angle of the triangle opposite side a "A." Label the angle opposite side b "B," and label the angle opposite side c "C."
- 3
  Measure the lengths of sides a, b and c using a ruler. For example, side a = 3 inches; side b = 4 inches; side c = 6.77 inches.
- 4
  Find the sum of the squared lengths of sides a and b using a scientific calculator. For example, (3 x 3) + (4 x 4) = 25.
- 5
  Find the difference of your answer and the square of the length of side c using a scientific calculator. For example, 25 - (6.77 x 6.77) = -20.8329.
- 6
  Divide your answer by the product of 2 and the lengths of sides a and b using a scientific calculator. For example, -20.8329/(2 x 3 x 4) = -0.8680375.
- 7
  Take the arc cosine of your answer using a scientific calculator. The arc cosine function on a scientific calculator is indicated by a button labeled "ARCOS" or "COS^-1." For example, arc cos (-0.8680375) = 150.231. The measurement of angle C according to the law of cosines is approximately 150 degrees.
- 8
  Take the sine of your answer using your scientific calculator. For example, sin (150.231) = 0.496498.
- 9
  Find the product of your answer and the length of side b. For example, 0.496498 x 4 = 1.98599.
- 10
  Divide your answer by the length of side c. For example, 1.98599/6.77 = 0.293352.
- 11
  Take the arc sine of your answer. The arc sine function on a scientific calculator is indicated by a button labeled "ARCSIN" or "SIN^-1." For example, arcsin (0.2933522) = 17.0587. The measurement of angle B according to the law of sines is approximately 17 degrees.
- 12
  Find the difference of the sum of the measurements of angles B and C and 180. For example, 180 - (150 + 17) = 13. The measurement of angle A is approximately 13 degrees.

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