Question

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Answer 1

`\displaystyle{x = (\pi)/(12), (\pi)/(6)}`

Explanation:

We are given the trigonometric equation of:

`\displaystyle{\sin 4x = (√(3))/(2)}`

Let u = 4x then:

`\displaystyle{\sin u = (√(3))/(2)}\\\\\displaystyle{\arcsin (\sin u) = \arcsin \left((√(3))/(2)\right)}\\\\\displaystyle{u= \arcsin \left((√(3))/(2)\right)}`

Find a measurement that makes sin(u) = √3/2 true within [0, π) which are u = 60° (π/3) and u = 120° (2π/3).

`\displaystyle{u = (\pi)/(3), (2\pi)/(3)}`

Convert u-term back to 4x:

`\displaystyle{4x = (\pi)/(3), (2\pi)/(3)}`

Divide both sides by 4:

`\displaystyle{x = (\pi)/(12), (\pi)/(6)}`

The interval is given to be 0 ≤ 4x < π therefore the new interval is 0 ≤ x < π/4 and these solutions are valid since they are still in the interval.

Therefore:

`\displaystyle{x = (\pi)/(12), (\pi)/(6)}`