Cosine and Arccosine in a right triangle

Cosine and Arccosine. What are they and how to calculate them?

Definitions and calculations cosine and arccosine



  • Definitions

  • Definition 1: Cosine - In a right triangle the cosine of an angle represents the ratio of the side adjacent to the angle to the hypotenuse.
  • cos(∠A) = c/b
  • Cosine and arccosine

    Cosine and arccosine

  • Definition 2: Arccos (the arccosine) - This is the inverse of the standard cosine mathematical function. While the cosine function takes an angle and returns the corresponding ratio, the arccos takes a ratio and returns the corresponding angle.
  • Since cos and arccos are inverse functions, arccos(cos(∡A)) = ∡A; if we apply the function arccos to the equality above:
  • cos(∡A) = c/b => arccos(cos(∡A)) = arccos(c/b) <=>
  • ∡A = arccos(c/b) =>
  • arccos(c/b) = measure of angle ∡A

  • Calculations examples

  • Example 1: Calculating the cosine
  • In a right triangle that has the measure of the angle ∡B = 90°, the length of side b = 10 and the length of side c = 6, the cosine of angle ∠A is calculated as:
  • cos(∠A) = c/b = 6/10 = 3/5 = 0.6.
  • Example 2: Calculating the arccosine
  • In a right triangle that has the measure of angle ∡B = 90°, the length of side b = 10 and the length of side c = 6, the measure of angle ∡A is calculated as:
  • arccos(cos(∠A)) = measure of angle ∡A = arccos(c/b) = arccos(6/10) = arccos(3/5) = arccos(0.6) = 53.1301024° ≈ 53.13°.