VertexMath · MHF4U · Free worksheet

Trigonometric Functions Review

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

Core path ~55 min Full sheet ~1 h 45 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

Radians: $\pi\text{ rad}=180^\circ$ . Arc length $s=r\theta$ only works with $\theta$ in radians, and so does your calculator: set it to radian mode.
Exact values: find the related acute angle, then take the sign from the quadrant (CAST). $\cos(-x)=\cos x$ but $\sin(-x)=-\sin x$ .
$y=a\sin(k(x-d))+c$ : amplitude $|a|$ , period $\dfrac{2\pi}{|k|}$ , phase shift $d$ , midline $y=c$ . Factor $k$ out of the bracket before reading $d$ .
Reciprocals: $\csc x=\dfrac{1}{\sin x}$ , $\sec x=\dfrac{1}{\cos x}$ , $\cot x=\dfrac{\cos x}{\sin x}$ . Vertical asymptotes sit where the denominator is $0$ . $\csc x$ is not $\sin^{-1}x$ .
A calculator gives one angle. Use CAST to find the second solution in each period, then add multiples of the period for the general solution.

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

Core Skills (K)

Radians · Graphs · Transformations · Reciprocals · Equations

K1 Radians & Exact Values

a)
Convert $210^\circ$ to radians and $\dfrac{7\pi}{4}$ to degrees.
b)
Evaluate exactly: $\sin\dfrac{7\pi}{6}$ , $\cos\left(-\dfrac{\pi}{4}\right)$ , $\tan\dfrac{5\pi}{3}$ .
c)
For $\theta=\dfrac{17\pi}{4}$ , determine the principal angle and one negative coterminal angle.

Stuck? Peek at a hint

Place each angle in its quadrant first. The related acute angle gives the value, the quadrant gives the sign.

K2 Key Features

a)
For $y=\tan x$ , state the period, the domain and the equations of the vertical asymptotes.
b)
For $y=\cos x$ on $0\le x\le 2\pi$ , state the zeros and the coordinates of every maximum and minimum point.
c)
Determine the smallest positive value of $d$ such that $\cos x=\sin(x+d)$ for every real $x$ .

Stuck? Peek at a hint

Write $\tan x=\dfrac{\sin x}{\cos x}$ and ask where the denominator is $0$ .

K3 Transformations

Let $f(x)=-3\sin\left(2x-\dfrac{\pi}{3}\right)+1$ .

a)
Write $f$ in the form $y=a\sin(k(x-d))+c$ . State the amplitude, period, phase shift and the equation of the midline.
b)
State the range of $f$ and the coordinates of its first maximum point with $x>0$ .
c)
Determine an equation of the form $y=a\cos(k(x-d))+c$ for the graph below. Each grid square is 1 unit.

Stuck? Peek at a hint

Factor $k$ out of the bracket before reading the phase shift. For the graph, the midline is halfway between the maximum and the minimum.

K4 Reciprocal Functions

a)
Evaluate exactly: $\csc\dfrac{5\pi}{4}$ , $\sec\dfrac{2\pi}{3}$ , $\cot\dfrac{\pi}{6}$ .
b)
State the domain, range and period of $y=\sec x$ .
c)
State the period and the equations of the vertical asymptotes of $y=\csc(2x)$ .

Stuck? Peek at a hint

Rewrite each function as a reciprocal, such as $\csc x=\dfrac{1}{\sin x}$ . Asymptotes sit where that denominator is $0$ .

K5 Trig Equations

a)
Solve $2\sin x+\sqrt{3}=0$ for $0\le x\le 2\pi$ .
b)
Solve $2\cos^2x-\cos x-1=0$ for $0\le x\le 2\pi$ .
c)
Determine the general solution of $\csc x=-2$ .

Stuck? Peek at a hint

Isolate the trig ratio first; in (b), factor as a quadratic in $\cos x$ . Then find the related acute angle and use CAST to place every solution.

Real World (A)

Sinusoidal Models · Rates of Change · Secant in Context

A1 Bay of Fundy Tide

A tide table gives the depth of water $D$ , in metres, at a wharf on the Bay of Fundy, $t$ hours after midnight.

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

a)
Determine an equation of the form $D(t)=a\sin(k(t-d))+c$ that models the depth.
b)
A fishing boat needs at least 5 m of water to dock. Between midnight and noon, during what times can it not dock?
c)
The wharf floods when the depth reaches 15 m. Use your model to find when that first happens.

Stuck? Peek at a hint

Read the maximum and minimum depths first: the amplitude is half their difference and the midline is their average.

A2 Fall Fair Ferris Wheel

A Ferris wheel turns once every 8 minutes at a constant speed. A rider boards at the bottom, and their height, in metres, after $t$ minutes is $h(t)=22-20\cos\left(\frac{\pi}{4}t\right)$ .

a)
Determine the average rate of change of height from $t=0$ to $t=1$ , and from $t=1$ to $t=2$ . Round to two decimals.
b)
Estimate the instantaneous rate of change of height at $t=2$ , using the interval $2\le t\le 2.001$ .
c)
Estimate the instantaneous rate of change at $t=4$ , using $3.999\le t\le 4.001$ . A rider concludes that the wheel stopped at $t=4$ . Is that right?

Stuck? Peek at a hint

Average rate of change is $\dfrac{h(b)-h(a)}{b-a}$ . Put your calculator in radian mode first.

A3 Georgian Bay Lighthouse

A lighthouse on an island in Georgian Bay is 500 m from a straight shoreline. Let $\theta$ be the angle between the beam and the line from the lighthouse straight to the shore. When the beam hits the shore, its length is $L(\theta)=500\sec\theta$ metres.

a)
Evaluate $L\left(\frac{\pi}{3}\right)$ and $L\left(\frac{\pi}{4}\right)$ . Give the second to the nearest tenth of a metre.
b)
The beam turns once every 20 s, so $\theta=\frac{\pi}{10}t$ after $t$ seconds. For $0\le t<5$ , when is the beam 1000 m long?
c)
Evaluate $L$ at $t=5$ s. What happens to the lit spot on the shore at that moment, and for $5<t<15$ ?

Stuck? Peek at a hint

Write $\sec\theta=\dfrac{1}{\cos\theta}$ and use exact values.

Think It Through (T)

Error Analysis · Building Functions

T1 Spot the Flaw

A student solves $2\sin x\cos x=\cos x$ for $0\le x\le 2\pi$ :
"Divide both sides by $\cos x$ : $2\sin x=1$ , so $\sin x=\frac{1}{2}$ . The solutions are $x=\frac{\pi}{6}$ and $x=\frac{5\pi}{6}$ ."

a)
Show that $x=\frac{\pi}{2}$ satisfies the equation.
b)
Identify the student's error, then solve the equation correctly.
c)
Without solving again, how many solutions does the equation have for $0\le x\le 4\pi$ ? Explain.

Stuck? Peek at a hint

Move every term to one side and look for a common factor.

T2 Build It Backwards

Write an equation for each function. Many answers are possible.

a)
A sine function with maximum value $7$ , minimum value $-1$ , period $\pi$ , and a maximum at $x=\frac{\pi}{3}$ .
b)
The same graph as (a), written as a cosine function.
c)
A trig function with period $\frac{\pi}{2}$ , vertical asymptotes exactly at $x=\frac{n\pi}{2}$ ( $n\in\mathbb{Z}$ ), whose graph passes through $\left(\frac{\pi}{8},1\right)$ .

Stuck? Peek at a hint

Start with the amplitude and midline from the max and min. A sine graph reaches its maximum a quarter period after it crosses the midline going up.

Boss Level

Optional challenge

Squeezed Solutions

Let $k$ be a positive integer. Consider the equation $\sin(kx)=\frac{1}{2}$ for $0\le x\le 2\pi$ .

a)
For $k=3$ , list every solution.
b)
For $k=3$ , find the sum of all the solutions.
c)
For a general positive integer $k$ , find the number of solutions and their sum, in terms of $k$ .

Stuck? Peek at a hint

Let $u=kx$ . If $0\le x\le 2\pi$ , what interval does $u$ run over?

Answer key

Final answers only. Try first, then check.

Core Skills

a) $\dfrac{7\pi}{6}$ ; $315^\circ$
b) $-\dfrac{1}{2}$ ; $\dfrac{\sqrt{2}}{2}$ ; $-\sqrt{3}$
c) Principal angle $\dfrac{\pi}{4}$ ; e.g. $-\dfrac{7\pi}{4}$

a) Period $\pi$ ; $D=\left\{x\in\mathbb{R}\mid x\ne\frac{\pi}{2}+n\pi,\ n\in\mathbb{Z}\right\}$ ; asymptotes $x=\frac{\pi}{2}+n\pi$ , $n\in\mathbb{Z}$
b) Zeros $\frac{\pi}{2}$ , $\frac{3\pi}{2}$ ; maxima $(0,1)$ and $(2\pi,1)$ ; minimum $(\pi,-1)$
c) $d=\dfrac{\pi}{2}$

a) $f(x)=-3\sin\left(2\left(x-\frac{\pi}{6}\right)\right)+1$ ; amplitude $3$ , period $\pi$ , phase shift $\frac{\pi}{6}$ right, midline $y=1$
b) $R=[-2,4]$ ; $\left(\frac{11\pi}{12},4\right)$
c) $y=2\cos\left(\frac{\pi}{3}(x-1)\right)-1$

a) $-\sqrt{2}$ ; $-2$ ; $\sqrt{3}$
b) $D=\left\{x\in\mathbb{R}\mid x\ne\frac{\pi}{2}+n\pi,\ n\in\mathbb{Z}\right\}$ ; $R=(-\infty,-1]\cup[1,\infty)$ ; period $2\pi$
c) Period $\pi$ ; asymptotes $x=\frac{n\pi}{2}$ , $n\in\mathbb{Z}$

a) $x=\dfrac{4\pi}{3},\ \dfrac{5\pi}{3}$
b) $x=0,\ \dfrac{2\pi}{3},\ \dfrac{4\pi}{3},\ 2\pi$
c) $x=\dfrac{7\pi}{6}+2n\pi$ or $x=\dfrac{11\pi}{6}+2n\pi$ , $n\in\mathbb{Z}$

Real World

a) $D(t)=6\sin\left(\frac{\pi}{6}t\right)+8$
b) From 7:00 a.m. to 11:00 a.m. ( $D(t)<5$ for $7<t<11$ )
c) Never: $D(t)=15$ needs $\sin\left(\frac{\pi}{6}t\right)=\frac{7}{6}>1$ . The model's maximum depth is 14 m, so the wharf does not flood.

a) $5.86$ m/min; $14.14$ m/min
b) About $15.71$ m/min
c) $0$ m/min. No: the wheel keeps turning at the same speed. At the top (42 m) the rider moves horizontally, so only their height stops changing for an instant.

a) $1000$ m; $500\sqrt{2}\approx 707.1$ m
b) $t=\frac{10}{3}\approx 3.33$ s
c) $L$ is undefined ( $\sec\frac{\pi}{2}$ is a vertical asymptote): the beam is parallel to the shore and never reaches it. For $5<t<15$ the beam points away from shore ( $L<0$ has no meaning), so there is no lit spot.

Think It Through

a) $\cos\frac{\pi}{2}=0$ , so both sides equal $0$ .
b) Dividing by $\cos x$ assumes $\cos x\ne0$ . Factor: $\cos x(2\sin x-1)=0$ , so $x=\frac{\pi}{6},\ \frac{\pi}{2},\ \frac{5\pi}{6},\ \frac{3\pi}{2}$ .
c) $8$ : both factors repeat every $2\pi$ , so each of the 4 solutions has one copy in $(2\pi,4\pi]$ , and $0$ , $2\pi$ , $4\pi$ are not solutions.

a) e.g. $y=4\sin\left(2\left(x-\frac{\pi}{12}\right)\right)+3$
b) e.g. $y=4\cos\left(2\left(x-\frac{\pi}{3}\right)\right)+3$
c) e.g. $y=\cot(2x)$

Boss Level

Boss

a) $x=\frac{\pi}{18},\ \frac{5\pi}{18},\ \frac{13\pi}{18},\ \frac{17\pi}{18},\ \frac{25\pi}{18},\ \frac{29\pi}{18}$
b) $5\pi$
c) $2k$ solutions; sum $(2k-1)\pi$

Free cheat sheet

Trigonometric 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 testedSolve $2\sin^2x-\sin x-1=0$ for $0\le x\le2\pi$ . (card 6)

1Radians and exact values

$\pi\text{ rad}=180^\circ\qquad s=r\theta$ ( $\theta$ in radians)

UseFind the related acute angle, then take the sign from CAST. $\cos(-x)=\cos x$ , $\sin(-x)=-\sin x$ . Coterminal: $\theta\pm2\pi$ .

Example $5\pi/4$ is in QIII with related angle $\pi/4$ : $\sin(5\pi/4)=-\sqrt2/2$ .

Trap $\cos(-\pi/4)=+\sqrt2/2$ : a negative angle does not make every ratio negative.

2Graphs of $\sin$ , $\cos$ , $\tan$

Use $\sin$ , $\cos$ : period $2\pi$ , range $[-1,1]$ . $\tan$ : period $\pi$ , zeros at $n\pi$ , asymptotes where $\cos x=0$ ( $x=\pi/2+n\pi$ ). Cofunction: $\sin(\pi/2-x)=\cos x$ .

Trap $\tan x$ has period $\pi$ , and its asymptotes are where $\cos x=0$ , not at its zeros.

3Transformations

$y=a\sin[k(x-d)]+c\qquad T=\dfrac{2\pi}{|k|}$

UseAmplitude $|a|$ , midline $y=c$ . Factor $k$ out first. For $a>0$ , sine reaches its maximum a quarter period after $d$ .

Trap $\sin(2x-\pi/3)=\sin[2(x-\pi/6)]$ : the shift is $\pi/6$ , not $\pi/3$ .

4Modelling

Use $a=(\max-\min)/2$ , $c=(\max+\min)/2$ , $k=2\pi/T$ . With a maximum at $t=d$ , use cosine.

ExampleHigh tide $5$ m at $2$ h, low $1$ m, every $12$ h: $h=2\cos[\pi/6\,(t-2)]+3$ .

TrapHigh to low tide is half a period: $6$ h means $T=12$ h.

5Reciprocal functions

$\csc x=\dfrac{1}{\sin x}\quad\sec x=\dfrac{1}{\cos x}$

$\cot x=\dfrac{\cos x}{\sin x}$

UseAsymptotes where the denominator is $0$ . $\csc$ , $\sec$ : period $2\pi$ ; $\cot$ : period $\pi$ . $\csc(kx)$ has asymptotes at $x=n\pi/k$ .

Trap $\csc x$ is $1/\sin x$ , not $\sin^{-1}x$ .

6Solving equations

UseRelated angle, then every quadrant CAST allows; general solution $+2\pi n$ ( $+\pi n$ for $\tan$ ). Factor quadratic forms; never divide by $\cos x$ . Convert $\csc$ , $\sec$ , $\cot$ first.

Example $(2\sin x+1)(\sin x-1)=0$ : $x=\pi/2,\ 7\pi/6,\ 11\pi/6$ .

Trap $\sin3x=1/2$ on $[0,2\pi]$ : $3x$ runs over $[0,6\pi]$ , so there are $6$ solutions, not $2$ .

7Rates of change

average rate $=\dfrac{f(b)-f(a)}{b-a}$

UseInstantaneous: shrink the interval. $\sin x$ changes fastest at its zeros and has rate $0$ at a max or min.

Example $\sin x$ from $0$ to $\pi/2$ : $(1-0)/(\pi/2)=2/\pi\approx0.64$ .

TrapA rate of $0$ at the top of a Ferris wheel means height isn't changing at that instant, not that the wheel stopped.

Which period?

\sin

\cos

\csc

\sec

2\pi/|k|

\tan

\cot

\pi/|k|

Data or context max to next max

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Core Skills (K)

K1 Radians & Exact Values

K2 Key Features

K3 Transformations

K4 Reciprocal Functions

K5 Trig Equations

Real World (A)

A1 Bay of Fundy Tide

A2 Fall Fair Ferris Wheel

A3 Georgian Bay Lighthouse

Think It Through (T)

T1 Spot the Flaw

T2 Build It Backwards

Boss Level

Squeezed Solutions

Answer key

Core Skills

Real World

Think It Through

Boss Level

Trigonometric Functions Cheat Sheet

Unit Summary

Build your own

Phone card

1Radians and exact values

2Graphs of sin⁡\sinsin, cos⁡\coscos, tan⁡\tantan

3Transformations

4Modelling

5Reciprocal functions

6Solving equations

7Rates of change

Which period?

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What is VertexMath?

2Graphs of $\sin$ , $\cos$ , $\tan$