Graphs of trigonometric functions · Mathematics 5th Year HL

The graphs of sin θ and cos θ

Take a point going around the unit circle. Its height is $\sin\theta$ and its horizontal position is $\cos\theta$. After one full turn ($360^\circ$) it is back where it started, so the values repeat. Plotting $\sin\theta$ against $\theta$ gives the wave below.

Both graphs have the same shape. The cosine graph is the sine graph shifted $90^\circ$ to the left. Three words describe any wave:

For $\sin\theta$ and $\cos\theta$: amplitude $1$, period $360^\circ$ ($2\pi$ radians), range $-1 \le y \le 1$.

Multiplying by a number $b$ stretches the wave vertically: $y = b\sin\theta$ has amplitude $|b|$ and range $-|b| \le y \le |b|$. The period does not change. For example, $y = 3\sin\theta$ has amplitude $3$, period $360^\circ$ and range $[-3, 3]$.

Multiplying the angle by $c$ squeezes the wave horizontally: $y = \sin c\theta$ completes $c$ cycles in $360^\circ$, so

$$\text{period} = \frac{360^\circ}{c} \quad \left(\text{or } \frac{2\pi}{c} \text{ in radians}\right).$$

Take $y = \cos 2\theta$. It has amplitude $1$ and period $180^\circ$. At $\theta = 0^\circ$ it is $\cos 0^\circ = 1$ (a maximum). At $\theta = 45^\circ$ it is $\cos 90^\circ = 0$.

Practice★☆☆

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

Solve

Practice★☆☆

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

Solve

The graph of tan θ

Recall that $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. Wherever $\cos\theta = 0$ we would be dividing by zero, so $\tan\theta$ does not exist there. This happens at $\theta = 90^\circ$ and $270^\circ$ in the interval $0^\circ \le \theta \le 360^\circ$. The graph shoots off to $\pm\infty$ near these values. The vertical lines $\theta = 90^\circ$ and $\theta = 270^\circ$ are the asymptotes: the graph gets closer and closer to them but never touches them.

The period of $\tan\theta$ is $180^\circ$, not $360^\circ$. The curve repeats after half a turn.
There is no maximum or minimum: the range is all real numbers.
$\tan\theta = 0$ where $\sin\theta = 0$, so for $0^\circ \le \theta \le 360^\circ$ the zeros are $0^\circ$, $180^\circ$ and $360^\circ$.
The graph always rises between asymptotes, so it has no amplitude.

Practice★☆☆

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…

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 $h(\theta)=0$.

Solve

The graphs of a + b sin cθ and a + b cos cθ

Put the three changes together. For $y = a + b\sin c\theta$ (and the same for cosine):

Why? The $\sin$ part always lies between $-1$ and $1$. Multiplying by $b$ makes it lie between $-|b|$ and $|b|$, and adding $a$ shifts everything up by $a$. If $b$ is negative, the wave is flipped upside down (it starts by going down).

Practice★★☆

Let $f(\theta) = 2 + 3\sin 2\theta$. a) State the amplitude, period, maximum and minimum values. b) Find the values of $\theta$ in $[0^\circ, 360^\circ]$ for which…

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

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

Solve

Practice★★☆

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…

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 $\theta = 30^\circ$.

Solve

Solving equations from the graph

To solve $f(\theta) = k$ for a trigonometric function $f$, find the angles where the horizontal line $y = k$ meets the graph. The graph shows how many solutions to expect. Algebra finds their values.

Solve 1 + 2cos 2θ = 0 for 0° ≤ θ ≤ 360°

Step 1: isolate the trig function. $2\cos 2\theta = -1$, so $\cos 2\theta = -\tfrac12$.

Step 2: widen the interval. If $0^\circ \le \theta \le 360^\circ$, then $0^\circ \le 2\theta \le 720^\circ$. We need every angle $2\theta$ in this interval, which is two full turns.

Step 3: use the CAST diagram. The reference angle is $\cos^{-1}\tfrac12 = 60^\circ$. Cosine is negative in the 2nd and 3rd quadrants, so $2\theta = 180^\circ - 60^\circ = 120^\circ$ and $2\theta = 180^\circ + 60^\circ = 240^\circ$.

Step 4: add 360° for the second turn. $2\theta = 120^\circ + 360^\circ = 480^\circ$ and $2\theta = 240^\circ + 360^\circ = 600^\circ$.

Step 5: divide by 2. $\theta = 60^\circ,\ 120^\circ,\ 240^\circ,\ 300^\circ$. There are four solutions, two for each of the two cycles.

Solve 3 + 4 sin 2θ = 1 using CAST

Solve $3 + 4\sin 2\theta = 1$ for $0^\circ \le \theta \le 360^\circ$, correct to the nearest $0.1^\circ$.

Isolate. $4\sin 2\theta = -2$, so $\sin 2\theta = -\tfrac12$. The reference angle is $\sin^{-1}\left(\tfrac12\right) = 30^\circ$.

Use CAST. Sine is negative in the 3rd and 4th quadrants, so $2\theta = 180^\circ + 30^\circ = 210^\circ$ and $2\theta = 360^\circ - 30^\circ = 330^\circ$. Add $360^\circ$ for the second turn: $570^\circ$ and $690^\circ$.

Divide by 2. $\theta = 105.0^\circ,\ 165.0^\circ,\ 285.0^\circ,\ 345.0^\circ$ (to the nearest $0.1^\circ$). Four solutions, as the counting rule predicts for two periods.

Using k. The equation $f(\theta)=k$ has the most solutions when $k$ lies strictly between the minimum and maximum of $f$, so compare $k$ with $a - |b|$ and $a + |b|$.

Practice★★★

Let $f(\theta) = 1 - 4\sin 3\theta$ for $0^\circ \le \theta \le 360^\circ$. a) State the amplitude, period, and range of $f$. b) Find the number of solutions of…

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

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

Solve

Modelling with waves

Anything that repeats smoothly can be modelled by $h = a + b\cos ct$ or $h = a + b\sin ct$. The middle line $a$ is the average value, $|b|$ is the amplitude and $c$ is chosen so that the period fits the situation: $c = \dfrac{360^\circ}{\text{period}}$.

The angle $ct$ is in degrees here, so make sure your calculator is in degree mode.

Practice★★☆

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

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

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

Solve

Practice★★★

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…

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 $t = 0$.

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

Solve

Where two graphs meet

To find where two trigonometric graphs cross, set the two expressions equal and solve. The two sides often involve different multiples of the angle, such as $\theta$ and $2\theta$. Rewrite them with the same angle using a formula from the Formulae and Tables booklet, for example

$$\cos 2\theta = 2\cos^2\theta - 1 \quad\text{or}\quad \cos 2\theta = 1 - 2\sin^2\theta, \qquad \cos^2\theta + \sin^2\theta = 1.$$

Practice★★★

On the same axes, consider $y = 2\sin\theta$ and $y = 1 + \cos 2\theta$ for $0^\circ \le \theta \le 360^\circ$. a) Show that the intersection points satisfy…

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

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

Solve

Let's practice! 3 medium exercises in this section

★★☆ Let $f(\theta) = 2 + 3\sin 2\theta$. a) State the amplitude, period, maximum and minimum…★★☆ The height $h$ metres of the tide in Dún Laoghaire harbour at $t$ hours after midnight is…★★☆ The graph of $y = a + b\cos c\theta$ has maximum value 7, minimum value $-1$ and period…

Summary

For $y = a + b\sin c\theta$ or $y = a + b\cos c\theta$: amplitude $|b|$, period $\dfrac{360^\circ}{c}$, middle line $y = a$, range $a - |b| \le y \le a + |b|$.
$\sin$ and $\cos$ have period $360^\circ$ and range $[-1, 1]$. $\tan\theta$ has period $180^\circ$, range all of $\mathbb{R}$, asymptotes where $\cos\theta = 0$ ($90^\circ$ and $270^\circ$ in $0^\circ$ to $360^\circ$) and zeros at $0^\circ$, $180^\circ$, $360^\circ$.
To find $a$ and $b$ from a graph: $a$ is the average of the maximum and the minimum, and $b$ is half their difference.
To solve an equation in $c\theta$, widen the interval to $c \times 360^\circ$, use the CAST diagram, then divide by $c$.
The tide in Dún Laoghaire: $h(t) = 3 + 1.5\cos(30t)^\circ$ has period $12$ hours, a maximum of $4.5$ m at midnight and a minimum of $1.5$ m at 6.00 a.m. Reading $a$, $b$ and $c$ gave all of this at once.