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

  • The cosine of supplementary angles

  • Definition: Two angles are supplementary if the sum of their measures equals 180 degrees (π radians).
  • Example: An angle of 120° is supplementary to the one of 60°, because 120° + 60° = 180°.
  • Supplementary angles cosine formula:

cos(180° - θ) = - cos(θ)


  • How do we prove this formula?
  • On the system of orthogonal axes xOy we have the circle with center O and radius r.
  • We choose the point A on the circle so that the measure of the angle made by the segment OA with the axis Ox is equal to θ. C is the projection of point A on the axis Ox, i.e. AC is perpendicular to Ox. => In the right triangle △AOC, the measure of the angle ∡AOC is m(∡AOC) = θ.
  • The cosine of supplementary angles

    The cosine of supplementary angles

  • We choose point B on the circle so that the measure of the angle made by the segment OB with the axis Ox is equal to (180° - θ). D is the projection of point B on the Ox axis, i.e. BD is perpendicular to Ox. => In the right triangle △BOD, the measure of the angle ∡BOD is m(∡BOD) = θ.
  • Right triangles △AOC and △BOD are congruent (angle, θ, hypotenuse, r) => OC congruent to OD, OC ≅ OD.
  • The cosines of the angles ∡AOC and ∡BOC are:

  • cos(∡AOC) = cos(θ) = OC/r
  • cos(∡BOC) = cos(180° - θ) = OD/r

    • But OD = - OC (they have the same length, being congruent segments, but are in different quadrants in terms of Ox).


    • => cos(∡BOC) = cos(180° - θ) = - OC/r = - cos(θ)
    • => cos(180° - θ) = - cos(θ)

    • 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:
    • Cosine and arccosine

      Cosine and arccosine


    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°.
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