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Fofino [41]
2 years ago
11

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

Mathematics
1 answer:
Alekssandra [29.7K]2 years ago
3 0

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

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