Sine and Arcsine in a Right Triangle

• Definitions

• Definition 1: The sine - In a right triangle the sine of an angle represents the ratio of the side opposite that angle to the hypotenuse.
• sin(∠A) = ^a/_b
• 

  Sine and arcsine

• Definition 2: Arcsin (the arcsine) - This is the inverse of the standard sine mathematical function. While the sine function takes an angle and returns the corresponding ratio, the arcsin takes a ratio and returns the corresponding angle.
• Since sin and arcsin are inverse functions, arcsin(sin(∡A)) = ∡A. If we apply the function arcsin to the equality above:
• sin(∡A) = ^a/_b => arcsin(sin(∡A)) = arcsin(^a/_b) <=>
• ∡A = arcsin(^a/_b) =>
• arcsin(^a/_b) = measure of angle ∡A

• The sine of supplementary angles

• Definition: Two angles are supplementary if the sum of their measures equals 180 degrees (π radians).
• Example: An angle of 110° is supplementary to the one of 70°, because 110° + 70° = 180°.
• Supplementary angles sine formula:

sin(180° - θ) = sin(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 sine 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) => AC congruent to BD, AC ≅ BD.
• The sines of the angles ∡AOC and ∡BOC are:

But BD = AC (they have the same length, being congruent segments and are in similar quadrants in terms of Ox).

• => sin(∡BOC) = sin(180° - θ) = ^AC/_r = sin(θ)
• => sin(180° - θ) = sin(θ)

• Calculations examples

• Example 1: Calculating the sine
• In a right triangle that has the measure of the angle ∡B = 90°, the length of side b = 10 (the side opposite the angle ∡B) and the length of side a = 4 (the side opposite the angle ∡A), the sine of angle ∠A is calculated as:
• 

  Sine and arcsine

sin(∠A) = ^a/_b = ⁴/₁₀ = ²/₅ = 0.4

• Example 2: Calculating the arcsine
• In a right triangle that has the measure of angle ∡B = 90°, the length of side b = 10 (the side opposite the angle ∡B) and the length of side a = 4 (the side opposite the angle ∡A), the measure of angle ∡A is calculated as:
• arcsin(sin(∠A)) = measure of angle ∡A = arcsin(^a/_b) = arcsin(⁴/₁₀) = arcsin(²/₅) = arcsin(0.4) = 23.578178478202° ≈ 23.58°.