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

Answer:

a) $x_{1} = \frac{\pi}{3}\,rad$, $x_{2} = \frac{5\pi}{3}\,rad$, b) $x_{1} = \frac{\pi}{4}\,rad$, $x_{2} = \frac{3\pi}{4}\,rad$, $x_{3} = \frac{5\pi}{4}\,rad$, $x_{4} = \frac{7\pi}{4} \,rad$

Step-by-step 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) $\frac{1}{\cos x} = 2$

5) $\cos x = \frac{1}{2}$

6) $x = \cos^{-1} \frac{1}{2}$

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

$x_{1} = \frac{\pi}{3}\,rad$, $x_{2} = \frac{5\pi}{3}\,rad$

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

1) $6\cdot \cos^{2} x = 3$

2) $\cos^{2} x = \frac{1}{2}$

3) $\cos x = \pm\frac{\sqrt{2}}{2}$

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

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

$x_{1} = \frac{\pi}{4}\,rad$, $x_{2} = \frac{3\pi}{4}\,rad$, $x_{3} = \frac{5\pi}{4}\,rad$, $x_{4} = \frac{7\pi}{4} \,rad$