Question

Solve the following for 0 ≤ θ ≤ 720°

a) 3cosθ – 2 = 0

b) 2sin2θ – sinθ – 1 = 0
Solve the following for -2π ≤ θ ≤ 2π

a) 3tan2θ – 1 = 0

b) 12cos2θ – 6 = sinθ

Answer 1

Answer on Question #56476 – Math – Trigonometry

Solve the following for $0 \leq \theta \leq 720{}^{\circ}$

a) $3\cos\theta - 2 = 0$

**Solution**

for $0 \leq \theta \leq 720{}^{\circ}$

$3\cos\theta - 2 = 0$

$3\cos\theta = 2$

$\cos\theta = 2/3$

$\theta = \arccos(2/3) + 2\pi n, n = 0,1$, where $\arccos()$ is the inverse of cosine function.

b) $2\sin^2(\theta) - \sin(\theta) - 1 = 0$

**Solution**

for $0 \leq \theta \leq 720{}^{\circ}$

$2\sin^2(\theta) - \sin(\theta) - 1 = 0$

Substitution $\sin(\theta) = x$

$$2x^2 - x - 1 = 0$$ $$D = 1 + 4 \cdot 2 = 9$$ $$x_1 = \frac{1 + 3}{4} = 1, \quad \sin(\theta) = 1, \quad \theta = \frac{\pi}{2} + 2\pi n, \quad n = 0,1; \quad \theta = \frac{\pi}{2}; \quad \frac{5\pi}{2}$$ $$x_2 = \frac{1 - 3}{4} = -\frac{1}{2}, \quad \sin(\theta) = -\frac{1}{2}, \quad \theta = (-1)^{k + 1} \frac{\pi}{6} + n\pi, \quad n = 0,1,2,3$$ $$\theta = \frac{\pi}{6}; \quad \frac{5\pi}{6}; \quad \frac{7\pi}{6}; \quad \frac{11\pi}{6}; \quad \frac{13\pi}{6}; \quad \frac{17\pi}{6}; \quad \frac{19\pi}{6}; \quad \frac{23\pi}{6}$$

Solve the following for $-2\pi \leq \theta \leq 2\pi$

a) $3\tan(2\theta) - 1 = 0$

**Solution**

for $-2\pi \leq \theta \leq 2\pi$

$3\tan(2\theta) - 1 = 0$

$3\tan(2\theta) = 1$

$\tan(2\theta) = 1/3$

$2\theta = \arctan(1/3) + \pi n, \quad n = -1,0,1$

\theta = 1 / 2

\cdot

arctan(1/3) + \pi n / 2,

n = -2, -1, 0, 1, 2

b) $12\cos (2\theta) - 6 = \sin \theta$

Solution

for $-2\pi \leq \theta \leq 2\pi$

12cos(2θ) - 6 = sinθ

use $\sin^2\theta = \frac{1 - \cos(2\theta)}{2} \Rightarrow \cos (2\theta) = 1 - 2\sin^2 (\theta)$

12 $(1 - 2\sin^2 (\theta)) - 6 = \sin (\theta)$

$12 - 24\sin^2 (\theta) - 6 = \sin (\theta)$

$-24\sin^2 (\theta) - \sin (\theta) + 6 = 0$

$24\sin^2 (\theta) + \sin (\theta) - 6 = 0$

Substitution

$\sin (\theta) = x$

$24x^{2} + x - 6 = 0$

$D = 1 + 4\cdot 24\cdot 6 = 577$

$x_{1} = \frac{-1 + \sqrt{577}}{2\cdot 24} = \frac{-1 + \sqrt{577}}{48}$

$\sin (\theta) = \frac{-1 + \sqrt{577}}{48},$

$\theta = (-1)^n \arcsin \left( \frac{-1 + \sqrt{577}}{48} \right) + n\pi, n = -1, 0, 1$

$\theta = -\arcsin \left(\frac{-1 + \sqrt{577}}{48}\right) - \pi; -\arcsin \left(\frac{-1 + \sqrt{577}}{48}\right); \arcsin \left(\frac{-1 + \sqrt{577}}{48}\right); \arcsin \left(\frac{-1 + \sqrt{577}}{48}\right) + \pi,$

$x_{2} = \frac{-1 + \sqrt{577}}{2\cdot 24} = \frac{-1 - \sqrt{577}}{48},$

$\sin (\theta) = \frac{-1 - \sqrt{577}}{48},$

$\theta = (-1)^n \arcsin \left( \frac{-1 - \sqrt{577}}{48} \right) + n\pi, n = -1, 0, 1$

$\theta = -\arcsin \left(\frac{-1 - \sqrt{577}}{48}\right) - \pi; -\arcsin \left(\frac{-1 - \sqrt{577}}{48}\right); \arcsin \left(\frac{-1 - \sqrt{577}}{48}\right); \arcsin \left(\frac{-1 - \sqrt{577}}{48}\right) + \pi.$

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