Principal solution for sin x = ½
sin x = ½
Here sin is positive,
We know that
sin is positive in 1st and 2nd quadrant
Value in 1st Quadrant = 30°
Value in 2nd Quadrant = 180° – 30° = 150°
So, Principal solutions are
x = 30° = 30° × π/180 = π/6
x = 150° = 150° × π/180 = 5π/6
Thus, Principal solutions are
π/6 & 5π/6
Principal solution for cos x = –1/√2
cos x = –1/√2
Here cos is negative,
We know that
cos is negative in 2nd and 3rd quadrant
Here, θ = 45°
Value in 2nd Quadrant = 180° – 45° = 135°
Value in 3rd Quadrant = 180° + 45° = 225°
So, Principal solutions are
x = 135° = 135° × π/180 = 3π/4
x = 225° = 225° × π/180 = 5π/4
Thus, Principal solutions are
3π/4 & 5π/4
Principal solution for tan x = –1
tan x = –1
Here tan is negative,
We know that
tan is negative in 2nd and 4th quadrant
Here, θ = 45°
Value in 2nd Quadrant = 180° – 45° = 135°
Value in 4th Quadrant = 360° – 45° = 315°
So, Principal solutions are
x = 135° = 135° × π/180 = 3π/4
x = 315° = 315° × π/180 = 7π/4
Thus, Principal solutions are
3π/4 & 7π/4
