A student attempts to solve $$2\cos x = 3\sin x$$ in the interval $-180^{\circ}\leqslant x\leqslant 180^{\circ}$ as follows: $$\tan x = \dfrac{3}{2}\Rightarrow x = 56.3^{\circ}, -123.7^{\circ}$$ Identify the mistake made by the student and solve the equation.
It should be $\tan x = \dfrac{2}{3}$ <br> $x = -146.3^{\circ}, 33.7^{\circ}$
Solve, in the interval $0\leqslant\theta\leqslant360^{\circ}$: $$\tan(\theta - 45^{\circ})=-1$$
$0, 180^{\circ}, 360^{\circ}$
The height of a balloon is given by $$h = \frac{1}{2}\sin(1200t)^{\circ}$$ where $h$ is the height in metres relative to its original position, and $t$ is the time in minutes. Find the time, in seconds, at which the balloon has first fallen $0.2$ metres.
$10.2$
Solve, in the interval $0\leqslant\theta\leqslant180^{\circ}$: $$4\sin3\theta = 3$$ Round your answers to the nearest degree.
$16^{\circ}, 44^{\circ}, 136^{\circ}, 164^{\circ}$
Solve, in the interval $0\leqslant\theta\leqslant360^{\circ}$: $$2\tan\frac{1}{3}\theta = 5$$
$205^{\circ}$
Solve, in the interval $-180^{\circ}\leqslant x\leqslant180^{\circ}$: $$3\sin3x=2\cos3x$$
$-169^{\circ}, -109^{\circ}, -49^{\circ}, 11^{\circ}, 71^{\circ}, 131^{\circ}$
Solve, in the interval $-360^{\circ}\leqslant x\leqslant360^{\circ}$: $$\sqrt{3}\sin(x-60^{\circ})+\cos(x-60^{\circ})=0$$
$-330^{\circ}, -150^{\circ}, 30^{\circ}, 210^{\circ}$
Solve,in the interval $0\leqslant\theta\leqslant2\pi$: $$\tan^2\left(\theta-\frac{\pi}{4}\right)=1$$
$0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2}, 2\pi$
Solve, in the interval $0\leqslant\theta\leqslant360^{\circ}$: $$3\sin^2\theta+\sin\theta = 0$$
$0^{\circ}, 180^{\circ}, 199^{\circ}, 341^{\circ}, 360^{\circ}$
A pendulum is modelled by $$\theta = 0.02\cos(20t)$$ where $\theta$ is its angle of displacement relative to the vertical and $t$ is the time in minutes. All angles are measured in radians. - Find the exact time taken, in minutes, for the pendulum to return to its starting position.
- Find all the times in the first minute that the pendulum has a displacement of $0.015$ radians.
- $\dfrac{\pi}{10}$
- $0.0361,0.278,0.350,0.592,0.664,0.906,0.979$
Solve, in the interval $0\leqslant\theta\leqslant2\pi$: $$2\sin^2\theta=3(1-\cos\theta)$$
$0, \dfrac{\pi}{3}, \dfrac{5\pi}{3}, 2\pi$
Solve, in the interval $0\leqslant\theta\leqslant360^{\circ}$: $$4(\sin^2\theta-\cos\theta)=3-2\cos\theta$$
$72^{\circ}, 144^{\circ}, 216^{\circ}, 288^{\circ}$
Solve, in the interval $-\pi \leqslant x \leqslant \pi$: $$\cos\left(2x-\dfrac{2\pi}{3}\right) = 0$$
$-\dfrac{11\pi}{12}, -\dfrac{5\pi}{12}, \dfrac{\pi}{12}, \dfrac{7\pi}{12}$
Solve, in the interval $0 \leqslant x \leqslant 2\pi$: $$\sin\left(2x+\dfrac{\pi}{6}\right)=\dfrac{\sqrt{3}}{2}$$
$\dfrac{\pi}{12}, \dfrac{\pi}{4}, \dfrac{13\pi}{12}, \dfrac{5\pi}{4}$
Solve, in the interval $-180^{\circ} \leqslant x \leqslant 180^{\circ}$: $$\sin(3x+45^{\circ}) = 0.5$$ Round your answers to the nearest degree.
$-125^{\circ}, -85^{\circ}, -5^{\circ}, 35^{\circ}, 115^{\circ}, 155^{\circ}$
Solve, in the interval $-\pi \leqslant x \leqslant \pi$: $$3\tan\left(\frac{\pi}{2} - 2x\right) = \sqrt{3}$$
$-\dfrac{5\pi}{6},-\dfrac{\pi}{3},\dfrac{\pi}{6},\dfrac{2\pi}{3}$
Solve, in the interval $0\leqslant\theta\leqslant360^{\circ}$: $$2\cos^22\theta-5\cos2\theta+2=0$$
$30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}$
Solve, in the interval $0\leqslant\theta\leqslant180^{\circ}$: $$4\sin3\theta=\tan3\theta$$ Round your answers to the nearest degree.
$0^{\circ}, 25^{\circ}, 60^{\circ}, 94^{\circ}, 120^{\circ}, 145^{\circ}, 180^{\circ}$