Complementary angles

Suppose we have two angles α = 50 ⁰ and β = 40 ⁰; If we add them, it gives us 90 ⁰, therefore, we say that α and β “complement each other”, that is, they are complementary angles.

Definition

The complementary angles are those that when added together result in 90⁰ (or π / 2 rad).

complementary angles

α and β are complementary since 62⁰ + 28⁰ = 90⁰

If we have two that are consecutive, the uncommon sides of both form a right angle.

complementary angles2

Complementary angles in a right triangle

We know that the sum of the interior angles of any triangle is equal to 180⁰. Therefore, in a right triangle the two acute angles are complementary.

complementary angles3α + β = 90⁰

Trigonometric ratios of complementary angles

If α and β are complementary (α + β = 90⁰), then β = 90⁰ - α:

  • without (90 - α) = cos α  
  • cos (90 - α) = without α
  • tg (90 - α) = ctg α
  • csc (90 - α) = sec α
  • sec (90 - α) = csc α
  • ctg (90⁰ - α) = tg α