Question

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Find all solutions of 5−4 secx = −3 on the interval [0, 2π).

Answer 1

`\theta=(\pi )/(3)\\\\\theta=(5\pi )/(3)`

Explanation:

`\theta = x`

`5-4 \sec (\theta) = -3`

`-4 \sec (\theta) = -3-5`

`-4 \sec (\theta) = -8`

`(-4 \sec (\theta))/(-4) =(-8)/(-4)`

`\sec (\theta)=2`

Once

`\sec(\theta)=(1)/(\cos(\theta))`

`\sec (\theta)=2 \implies (1)/(\cos(\theta))=2 \implies \cos(\theta)=(1)/(2)`

The solutions for
`\cos(\theta)=(1)/(2)` are

`\theta=(\pi )/(3)+2\pi n, n \in \mathbb{Z}\\\\\theta=(5\pi )/(3)+2\pi n, n \in \mathbb{Z}`

But once the interval for the solution was given,

For
`[0, 2\pi)`

`\theta=(\pi )/(3)\\\\\theta=(5\pi )/(3)`

Answer 2

uterlly useless considering 11-34 is 3