VertexMath · MCR3U · Free worksheet

Sinusoidal Functions Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 55 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Amplitude $=\dfrac{\max-\min}{2}$ , always positive. Equation of the axis: $y=\dfrac{\max+\min}{2}$ , written as an equation.
For $y=a\sin(k(x-d))+c$ : amplitude $|a|$ , period $\dfrac{360^\circ}{|k|}$ , phase shift $d$ (right when $d>0$ ), axis $y=c$ .
Factor $k$ out first: $\sin(2x-60^\circ)=\sin(2(x-30^\circ))$ is a shift of $30^\circ$ right, not $60^\circ$ .
$\cos x=\sin(x+90^\circ)$ . A sine curve starts a cycle on its axis going up; a cosine curve starts at a maximum.
Solving $\sin\theta=$ value gives two angles per cycle. Find both, then add multiples of the period.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Periodic Functions · Sine & Cosine · Transformations · Equations from Graphs

K1 Read the Cycle

The graph shows a periodic function $f$ . Each grid square is 1 unit.

a)
State the period.
b)
State the amplitude and the equation of the axis.
c)
Determine $f(44)$ .

Stuck? Peek at a hint

A full cycle ends when the graph is back at the same height and moving in the same direction. Find two consecutive maximum points.

K2 Sine & Cosine Values

Do not use a calculator.

a)
Evaluate $\sin 240^\circ$ and $\cos 240^\circ$ exactly.
b)
Evaluate $\cos 135^\circ$ and $\sin 330^\circ$ exactly.
c)
For $y=\cos x$ , $0^\circ\le x\le 360^\circ$ , state the zeros, the maximum points and the minimum point.

Stuck? Peek at a hint

Find the reference angle, then use the quadrant for the sign: $\cos\theta$ is the $x$ -coordinate on the unit circle, $\sin\theta$ the $y$ -coordinate.

K3 Read the Equation

State the requested features of each function.

a)
$y=-3\sin(2(x-30^\circ))+1$ : amplitude, period, phase shift and equation of the axis.
b)
$y=4\cos\left(\frac{1}{3}x\right)-2$ : amplitude, period and range.
c)
$y=\sin(3x+90^\circ)$ : period and phase shift.

Stuck? Peek at a hint

Factor $k$ out of the bracket so it reads $k(x-d)$ . The period is $\dfrac{360^\circ}{|k|}$ .

K4 Key Points

Let $y=-2\sin(2(x+45^\circ))+3$ .

a)
Determine the five key points of one cycle, starting at $x=-45^\circ$ .
b)
Evaluate $y$ at $x=15^\circ$ exactly.
c)
Find all $x$ with $0^\circ\le x\le 360^\circ$ where $y$ reaches its maximum value.

Stuck? Peek at a hint

Start from the key points of $y=\sin x$ and map each $(x,y)$ to $\left(\frac{x}{k}+d,\ ay+c\right)$ .

K5 Equation from a Graph

The graph shows a sinusoidal function, with $x$ in degrees. Vertical grid lines are $30^\circ$ apart and horizontal grid lines are 1 unit apart.

a)
State the maximum, the minimum, the amplitude and the equation of the axis.
b)
Determine the period and the value of $k$ .
c)
Determine an equation of the form $y=a\sin(k(x-d))+c$ .

Stuck? Peek at a hint

For a sine equation, $d$ is where the curve crosses its axis going up, not where it peaks.

Real World (A)

Periodic Models · Solving with Models

A1 Heartbeat

At rest, Jordan's blood pressure rises and falls with each heartbeat, between 80 mmHg and 120 mmHg. Their heart beats 75 times per minute. Model the pressure $P$ , in mmHg, as a sinusoidal function of the time $t$ , in seconds.

a)
State the amplitude, the equation of the axis and the period, with units.
b)
The pressure is at its maximum at $t=0$ . Determine $P(t)$ as a cosine function, then evaluate $P(0.3)$ .
c)
During a workout, Jordan's heart rate climbs steadily from 75 to 150 beats per minute. Which parameter of your model changes, and what is its value at 150 beats per minute? Can one function of this form model the whole workout?

Stuck? Peek at a hint

75 beats per minute is a rate. The period is the time for one beat, in seconds.

A2 Ferris Wheel

A Ferris wheel at a summer fair in Toronto has a radius of 12 m, and its axle is 14 m above the ground. It turns once every 80 s. A rider boards at the lowest point at $t=0$ .

a)
Determine $h(t)$ , the rider's height in metres after $t$ seconds.
b)
Find the height at $t=30$ s, exactly and to one decimal place.
c)
Find the first two times the rider is 20 m above the ground.
d)
A poster says riders reach 28 m at the top. Use the model to find when the rider is 28 m high, and interpret the result.

Stuck? Peek at a hint

Find the lowest and highest heights first: they give $a$ and $c$ .

A3 Bay of Fundy Tides

The water depth $d$ at a dock on the Bay of Fundy, Nova Scotia, was recorded every 3 hours starting at midnight ( $t=0$ ).

$t\text{ (h)}$	$0$	$3$	$6$	$9$	$12$	$15$
$d\text{ (m)}$	$8$	$14$	$8$	$2$	$8$	$14$

a)
State the amplitude, the period and the equation of the axis, with units.
b)
Determine an equation for $d(t)$ using a cosine function.
c)
Find all times with $0\le t\le 12$ when the depth is exactly 5 m.
d)
A cargo boat needs at least 5 m of water for its whole 2-hour unloading at the dock. During the first 12 hours, at what times can it start unloading?

Stuck? Peek at a hint

Measure the period from one high tide to the next high tide.

Think It Through (T)

Error Analysis · Building Functions

T1 Spot the Flaw

A student converts $y=5\cos(2(x-10^\circ))+8$ into a sine function:
"Cosine is sine shifted by $90^\circ$ , so I subtract $90^\circ$ from the phase shift: $y=5\sin(2(x-100^\circ))+8$ ."

a)
Evaluate both functions at $x=10^\circ$ . Are they the same function?
b)
Identify the error and write a correct sine equation.
c)
A second student writes $y=-5\sin(2(x-55^\circ))+8$ . Show that this is also correct.

Stuck? Peek at a hint

Write $\cos\theta=\sin(\theta+90^\circ)$ with $\theta=2(x-10^\circ)$ , then factor out the 2.

T2 Build It Backwards

A sinusoidal function has a maximum value of 7, a minimum value of $-1$ , a period of $90^\circ$ , and a maximum at $x=30^\circ$ .

a)
Write its equation in the form $y=a\sin(k(x-d))+c$ with $a>0$ .
b)
Write the same function in the form $y=a\cos(k(x-d))+c$ with $a>0$ .
c)
A student says the value of $d$ in (a) is the only one possible. Explain why other values work, and give two of them.

Stuck? Peek at a hint

A sine curve reaches its maximum a quarter of a period after it crosses its axis going up.

Boss Level

Optional challenge

Counting Solutions

Let $f(x)=4\sin(3x)+2$ for $0^\circ\le x\le 360^\circ$ .

a)
Solve $f(x)=4$ .
b)
Without solving, find the sum of all solutions of $f(x)=k$ for any $k$ with $2<k<6$ .
c)
Find the value of $k$ for which $f(x)=k$ has exactly 7 solutions.

Stuck? Peek at a hint

Let $u=3x$ and find the interval that $u$ covers when $0^\circ\le x\le 360^\circ$ .

Answer key

Final answers only. Try first, then check.

Core Skills

a) $8$
b) Amplitude $3$ ; axis $y=4$
c) $6$

a) $\sin 240^\circ=-\dfrac{\sqrt{3}}{2}$ ; $\cos 240^\circ=-\dfrac{1}{2}$
b) $\cos 135^\circ=-\dfrac{\sqrt{2}}{2}$ ; $\sin 330^\circ=-\dfrac{1}{2}$
c) Zeros $90^\circ$ , $270^\circ$ ; maximum points $(0^\circ,1)$ , $(360^\circ,1)$ ; minimum point $(180^\circ,-1)$

a) Amplitude $3$ ; period $180^\circ$ ; $30^\circ$ right; axis $y=1$
b) Amplitude $4$ ; period $1080^\circ$ ; range $[-6,2]$
c) Period $120^\circ$ ; $30^\circ$ left

a) $(-45^\circ,3)$ , $(0^\circ,1)$ , $(45^\circ,3)$ , $(90^\circ,5)$ , $(135^\circ,3)$
b) $3-\sqrt{3}$
c) $x=90^\circ$ and $x=270^\circ$

a) Max $5$ , min $1$ ; amplitude $2$ ; axis $y=3$
b) Period $120^\circ$ ; $k=3$
c) e.g. $y=2\sin(3(x-30^\circ))+3$ (any $d=30^\circ+120^\circ n$ works)

Real World

a) Amplitude $20$ mmHg; axis $P=100$ ; period $0.8$ s
b) $P(t)=20\cos(450t)+100$ ; $P(0.3)=100-10\sqrt{2}\approx 85.9$ mmHg
c) $k$ changes: $k=900$ (period $0.4$ s). No: $k$ is a constant in the model, so it only fits a stretch of time where the heart rate stays the same.

a) $h(t)=-12\cos(4.5t)+14$
b) $14+6\sqrt{2}\approx 22.5$ m
c) $t=\frac{80}{3}\approx 26.7$ s and $t=\frac{160}{3}\approx 53.3$ s
d) Never: $\cos(4.5t)=-\frac{7}{6}$ has no solution. The top of the wheel is $26$ m, so the poster's 28 m is wrong.

a) Amplitude $6$ m; period $12$ h; axis $d=8$ (m)
b) e.g. $d(t)=6\cos(30(t-3))+8$
c) $t=7$ h and $t=11$ h
d) Any start from $t=0$ to $t=5$ h, or from $t=11$ to $t=12$ h. A start between 5 h and 11 h runs into water shallower than 5 m.

Think It Through

a) $13$ and $8$ : not the same function
b) The $90^\circ$ shift acts on $2(x-10^\circ)$ , so it moves the graph $45^\circ$ , and to the left: $y=5\sin(2(x+35^\circ))+8$
c) $2(x-55^\circ)=2(x+35^\circ)-180^\circ$ , a half-period shift, and $\sin(\theta-180^\circ)=-\sin\theta$ , so $-5\sin(2(x-55^\circ))=5\sin(2(x+35^\circ))$ .

a) $y=4\sin(4(x-7.5^\circ))+3$
b) $y=4\cos(4(x-30^\circ))+3$
c) Shifting by a full period ( $90^\circ$ ) gives the same graph, so $d=7.5^\circ+90^\circ n$ works for every integer $n$ , e.g. $97.5^\circ$ and $-82.5^\circ$ .

Boss Level

Boss

a) $x=10^\circ, 50^\circ, 130^\circ, 170^\circ, 250^\circ, 290^\circ$
b) $900^\circ$
c) $k=2$

Free cheat sheet

Sinusoidal Functions Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

↓ Build-your-own PDF

Phone card

The essentials on one screen. Save it to your photos.

↓ Phone card (image)

Most testedA sinusoid has a maximum of $7$ at $x=30^\circ$ , a minimum of $-1$ and a period of $180^\circ$ . Write its equation. (card 5)

1Periodic functions

amplitude $=\dfrac{\max-\min}{2}$

axis $y=\dfrac{\max+\min}{2}$

UsePeriod: the horizontal length of one full cycle, back to the same height and heading the same way.

TrapCrossing the axis again is not a full cycle unless the curve is moving the same way as at the start.

TrapA tide between $2$ m and $10$ m has amplitude $4$ m (not $8$ ) and axis $y=6$ (not $5$ ).

2 $\sin x$ and $\cos x$

UseUnit circle point: $(\cos\theta,\sin\theta)$ . Both: amplitude $1$ , period $360^\circ$ , range $[-1,1]$ . Sine starts on the axis going up; cosine starts at a maximum. $\cos x=\sin(x+90^\circ)$ .

Trap $\cos120^\circ=-\cos60^\circ=-1/2$ : the related angle is $60^\circ$ , and cosine is negative in QII.

3Transformations

$y=a\sin[k(x-d)]+c$

period $=\dfrac{360^\circ}{|k|}$

UseAmplitude $|a|$ , phase shift $d$ , axis $y=c$ , range $[c-|a|,\ c+|a|]$ . Factor $k$ out first.

Trap $\sin(2x-60^\circ)=\sin[2(x-30^\circ)]$ : the shift is $30^\circ$ right, not $60^\circ$ .

Trap $k=1/3$ : period $360^\circ\div(1/3)=1080^\circ$ , not $120^\circ$ .

4Graphing with key points

UseFive key points per cycle, a quarter period apart. Sine: axis, max, axis, min, axis. Map each with $(x,y)\to(x/k+d,\ ay+c)$ .

Example $y=2\sin[3(x-10^\circ)]+1$ : $(0,0)\to(10^\circ,1)$ and $(90^\circ,1)\to(40^\circ,3)$ .

5Equation from a graph

Use $a=(\max-\min)/2$ , $c=(\max+\min)/2$ , $k=360^\circ/\text{period}$ . $d$ : a maximum for cosine, a rising axis crossing for sine. Keep the brackets.

ExampleMax $7$ at $30^\circ$ , min $-1$ , period $180^\circ$ : $y=4\cos[2(x-30^\circ)]+3$ .

TrapA period of $120^\circ$ gives $k=360^\circ/120^\circ=3$ , not $k=120$ .

6Modelling

UseFerris wheel: amplitude = radius, axis = hub height. Boarding at the bottom: $h=-a\cos(kt)+c$ . Use time units that match the period.

ExampleRadius $12$ m, hub $14$ m, one turn in $60$ s: $h=-12\cos(6t)+14$ .

TrapIn $h=-12\cos(6t)+14$ , $14$ m is the hub height; boarding is at $14-12=2$ m.

7Solving with a model

UseIsolate the trig function, find both angles in the cycle, then solve for $t$ . Add multiples of the period for later cycles.

Example $\cos(6t)=0.5$ : $6t=60^\circ$ or $300^\circ$ , so $t=10$ s or $50$ s in the first turn.

TrapThe calculator shows one angle; the second angle in the cycle is an answer too.

Sine or cosine?

Starts on the axis, going up

y=a\sin(kx)+c

Starts at a maximum

y=a\cos(kx)+c

Starts at a minimum

y=-a\cos(kx)+c

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Core Skills (K)

K1 Read the Cycle

K2 Sine & Cosine Values

K3 Read the Equation

K4 Key Points

K5 Equation from a Graph

Real World (A)

A1 Heartbeat

A2 Ferris Wheel

A3 Bay of Fundy Tides

Think It Through (T)

T1 Spot the Flaw

T2 Build It Backwards

Boss Level

Counting Solutions

Answer key

Core Skills

Real World

Think It Through

Boss Level

Sinusoidal Functions Cheat Sheet

Unit Summary

Build your own

Phone card

1Periodic functions

2sin⁡x\sin xsinx and cos⁡x\cos xcosx

3Transformations

4Graphing with key points

5Equation from a graph

6Modelling

7Solving with a model

Sine or cosine?

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What is VertexMath?

2 $\sin x$ and $\cos x$