Exercises: Graphs of trigonometric functions

For $f(\theta) = 3\sin\theta$, state the amplitude, the period and the range. Give the value of $f(90^\circ)$.

For $g(\theta) = \cos 2\theta$, state the amplitude and the period (in degrees). Find $g(0^\circ)$ and $g(45^\circ)$.

For $h(\theta) = \tan\theta$, state the period, the equations of the asymptotes for $0^\circ \le \theta \le 360^\circ$, and the values of $\theta$ in this interval for which $h(\theta)=0$.

Let $f(\theta) = 2 + 3\sin 2\theta$.

1. a) State the amplitude, period, maximum and minimum values.
2. b) Find the values of $\theta$ in $[0^\circ, 360^\circ]$ for which $f(\theta) = 5$.
3. c) Sketch the graph for $0^\circ\le\theta\le 360^\circ$, marking key points.

The height $h$ metres of the tide in Dún Laoghaire harbour at $t$ hours after midnight is modelled by $h(t) = 3 + 1.5\cos\left(30t\right)^\circ$.

1. a) Find the period of the tide in hours.
2. b) Find the maximum and minimum heights and the times of the first maximum and first minimum after midnight.
3. c) Find the first time after midnight at which $h = 3.75$.

The graph of $y = a + b\cos c\theta$ has maximum value 7, minimum value $-1$ and period $120^\circ$, with $b>0$ and $c>0$. Find $a$, $b$ and $c$, and find $y$ when $\theta = 30^\circ$.

Let $f(\theta) = 1 - 4\sin 3\theta$ for $0^\circ \le \theta \le 360^\circ$.

1. a) State the amplitude, period, and range of $f$.
2. b) Find the number of solutions of $f(\theta) = 0$ in the interval, and find all of them to the nearest $0.1^\circ$.
3. c) Find the range of values of $k$ for which $f(\theta)=k$ has exactly six solutions in the interval.

On the same axes, consider $y = 2\sin\theta$ and $y = 1 + \cos 2\theta$ for $0^\circ \le \theta \le 360^\circ$.

1. a) Show that the intersection points satisfy $2\cos^2\theta + \ldots$, i.e. show that $2\sin\theta = 1+\cos2\theta$ reduces to $\cos^2\theta = \sin\theta$.
2. b) Solve $\sin^2\theta + \sin\theta - 1 = 0$ and hence find the values of $\theta$ at the intersections, to the nearest $0.1^\circ$.
3. c) Hence state how many times the graphs intersect.

A Ferris wheel at a fair in Ennis has radius 12 m and its lowest point is 2 m above the ground. It completes one revolution every 90 seconds, and a passenger is at the lowest point when $t = 0$.

1. a) Show that the passenger's height is $H(t) = 14 - 12\cos(4t)^\circ$.
2. b) Find the times in the first revolution when the passenger is 17 m above the ground.
3. c) For what fraction of one revolution is the passenger higher than 20 m above the ground?