Answer:
The solutions of the equations are π/3 , 2π/3 , 4π/3 , 5π/3
Step-by-step explanation:
* Lets revise the four quadrant before solving the equation
# First quadrant the measure of all angles is between 0 and π/2
the measure of any angle is α
∴ All the angles are acute
∴ All the trigonometry functions of α are positive
# Second quadrant the measure of all angles is between π/2 and π
the measure of any angle is π - α
∴ All the angles are obtuse
∴ The value of sin(π - α) only is positive ⇒ sin(π - α) = sinα
# Third quadrant the measure of all angles is between π and 3π/2
the measure of any angle is π + α
∴ All the angles are reflex
∴ The value of tan(π + α) only is positive ⇒ tan(π + α) = tanα
# Fourth quadrant the measure of all angles is between 3π/2 and 2π
the measure of any angle is 2π - α
∴ All the angles are reflex
∴ The value of cos(2π - α) only is positive ⇒ cos(2π - α) = cosα
* Now lets solve the equation
∵ 4 sin²Ф - 3 = 0 ⇒ the domain is 0 ≤ Ф ≤ 2π
- Add 3 for both sides
∴ 4 sin²Ф = 3 ⇒ divide the both sides by 4
∴ sin²Ф = 3/4 ⇒ take square root for both sides
∴ √(sin²Ф) = √(3/4)
∴ sinФ = √3/2 OR sinФ = -√3/2
- When the value of sinФ is positive
∴ The angle Ф is on the first or second quadrant
- When the value of sinФ is negative
∴ The angle Ф is on the third or fourth quadrant
- We have four values of Ф because 0 ≤ Ф ≤ 2π
- Lets find the measure of the acute angle α
∵ sinα = √3/2
∴ α = sin^-1(√3/2) = π/3
- If Ф is on the first quadrant
∴ Ф = α = π/3
- If Ф is on the second quadrant
∴ Ф = π - α = π - π/3 = 2π/3
- If Ф is on the third quadrant
∴ Ф = π + α = π + π/3 = 4π/3
- If Ф is on the fourth quadrant
∴ Ф = 2π - α = 2π - π/3 = 5π/3
* The solutions of the equations are π/3 , 2π/3 , 4π/3 , 5π/3
