Question

Find the exact values of the solutions in the interval 0 <= x < 2pi of the following equations without using a calculator. a) 3sec(x) + 2 = 8

Answer 1

a)
`x_(1) = (\pi)/(3)\,rad`,
`x_(2) = (5\pi)/(3)\,rad`, b)
`x_(1) = (\pi)/(4)\,rad`,
`x_(2) = (3\pi)/(4)\,rad`,
`x_(3) = (5\pi)/(4)\,rad`,
`x_(4) = (7\pi)/(4) \,rad`

Explanation:

a) We proceed to solve the expression by algebraic and trigonometrical means:

1)
`3\cdot \sec x + 2 = 8`

2)
`3\cdot \sec x = 6`

3)
`\sec x = 2`

4)
`(1)/(\cos x) = 2`

5)
`\cos x = (1)/(2)`

6)
`x = \cos^(-1) (1)/(2)`

Cosine has positive values in first and fourth quadrants. Then, we have the following two solutions:

`x_(1) = (\pi)/(3)\,rad`,
`x_(2) = (5\pi)/(3)\,rad`

b) We proceed to solve the expression by algebraic and trigonometrical means:

1)
`6\cdot \cos^(2) x = 3`

2)
`\cos^(2) x = (1)/(2)`

3)
`\cos x = \pm(√(2))/(2)`

4)
`x = \cos^(-1) \left(\pm \frac{\sqrt {2}}{2} \right)`

There is one solution for each quadrant. That is to say:

`x_(1) = (\pi)/(4)\,rad`,
`x_(2) = (3\pi)/(4)\,rad`,
`x_(3) = (5\pi)/(4)\,rad`,
`x_(4) = (7\pi)/(4) \,rad`