calculate the value of a

Question

Calculate the values a that will satisfy the trigonometric equation for 0°≤a≤360°: 3sin a+2=0

Answer 1

The trigonometric equation is 3 sin(a) + 2 = 0.

sin(a) = - 2/3

sin(a) = sin[sin^-1(- 2/3)]

sin(a) = sin[- 41.81]

If sin(θ) = sin(α), then general solution θ = 180n + (-1)ⁿα, where n is an integer.

If θ = - 41.81 then θ = 180n + (-1)ⁿ[- 41.81] = 180n - (-1)ⁿ[- 41.81], where n is an integer.

If n = 0 then θ = 180(0) - (-1)⁽⁰⁾(41.81) = - 41.81.

If n = 1 then θ = 180(1) - (-1)⁽¹⁾(41.81) = 221.81.

If n = 2 then θ = 180(2) - (-1)⁽²⁾(41.81) = 318.19.

If n = 3 then θ = 180(3) - (-1)⁽³⁾(41.81) = 581.81.

The solutions in the interval [0, 360] are 221.81^o and 318.19^o.