Question

Just the answer please.
1: http://imgur.com/8Py19dX
2: http://imgur.com/VOWfj2t
3: http://imgur.com/UT0i1sN

Answer 1

Answer on Question #58927 – Math – Trigonometry

Question

Solve on the interval $[0, 2\pi)$:

$$4cscx + 6 = -2$$ $$\frac{\pi}{3}, \frac{5\pi}{3}$$ $$\frac{2\pi}{3}, \frac{4\pi}{3}$$ $$\frac{\pi}{6}, \frac{5\pi}{6}$$ $$\frac{7\pi}{6}, \frac{11\pi}{6}$$

Solution

$$4cscx + 6 = -2, 0 \leq x < 2\pi;$$ $$\frac{4}{\sin x} = -2 - 6;$$ $$\frac{4}{\sin x} = -8;$$ $$\sin x = -\frac{1}{2};$$ $$x = \frac{7\pi}{6} \text{ or } x = \frac{11\pi}{6}.$$

Answer: $\frac{7\pi}{6}, \frac{11\pi}{6}$.

Question

Solve on the interval $[0, 2\pi)$:

$$3 \sec x - 2 = 1$$

$$\frac{\pi}{3}, \frac{5\pi}{3}$$ $$\frac{\pi}{6}, \frac{5\pi}{6}$$ $$\frac{2\pi}{3}, \frac{4\pi}{3}$$

Solution

$$3 \sec x - 2 = 1, \quad 0 \leq x < 2\pi;$$ $$\frac{3}{\cos x} = 1 + 2;$$ $$\frac{3}{\cos x} = 3;$$ $$\cos(x) = 1;$$ $$x = 0.$$

Answer: 0.

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Question

Which value is a solution for the equation $\tan \frac{x}{2} = -1$?

$$\frac{3\pi}{4}$$ $$\frac{7\pi}{4}$$ $$\frac{5\pi}{4}$$ $$\frac{3\pi}{2}$$

Solution

$\tan \frac{x}{2} = -1;$

$\frac{x}{2} = -\frac{\pi}{4} + n\pi$, where $n$ is integer;

$$x = -\frac{\pi}{2} + 2n\pi.$$

If we take $n = 1$, then

$$x = -\frac{\pi}{2} + 2\pi = \frac{2 \cdot 2\pi - \pi}{2} = \frac{3\pi}{2}.$$

Answer: $\frac{3\pi}{2}$.