Secondary 5 Trigonometric Functions Worksheet

All 28 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Finding the Rule of a Cosine Function

   A cosine function reaches a maximum value of $11$ at $x=2$, and the very next minimum value it reaches is $-3$, at $x=8$. Find its rule in the form $f(x)=a\cos (b(x-h))+k$ with $a>0$ and $b>0$, using radians.

2. Q2Finding the Rule of a Sine Function

   A buoy bobs on the swell. Its height above mean sea level is $0$ m at $t=0$ s and rising, and it first reaches its greatest height, $1.8$ m, at $t=3$ s. The motion is sinusoidal.

   1. Find the rule $h(t)=a\sin (b(t-h_0))+k$ of this function.
   2. Find the buoy's height at $t=10$ s, to the nearest hundredth of a metre.
3. Q3Finding the Rule of a Tangent Function

   A tangent function has consecutive asymptotes at $x=\frac{\pi }{4}$ and $x=\frac{3\pi }{4}$. Its centre point is $(\frac{\pi }{2},3)$ and it passes through $(\frac{5\pi }{8},7)$. Find its rule in the form $f(x)=a\tan (b(x-h))+k$ with $b>0$, then compute $f\left(\frac{3\pi }{8}\right)$.

4. Q4Graphing a Cosine Function

   Consider $f(x)=3\cos (\frac{\pi }{4}(x-1))+2$.

   1. State the amplitude, the period, the equation of the midline, and the maximum and minimum values.
   2. Sketch the graph on the grid below for $-6\le x\le 6$, marking the maximum, minimum and midline points you used.

   A blank Cartesian grid for this question is on the printable PDF.

5. Q5Graphing a Sine Function

   Consider $f(x)=-4\sin (\frac{\pi }{3}(x-1))+1$.

   1. State the amplitude, the period, the midline, and the maximum and minimum values.
   2. Sketch the graph on the grid for $-6\le x\le 6$.
   3. Give the smallest value of $x$ greater than or equal to $1$ at which $f$ reaches its maximum.

   A blank Cartesian grid for this question is on the printable PDF.

6. Q6Graphing a Tangent Function

   Consider $f(x)=2\tan (\frac{\pi }{4}(x-1))+3$.

   1. Give the period of $f$.
   2. Give the equations of the two asymptotes on either side of $x=1$.
   3. Compute $f(0)$, $f(1)$ and $f(2)$.
   4. Sketch the branch of the graph lying between those two asymptotes.
7. Q7Solving Problems Involving Tangent Functions

   A lighthouse stands $200$ m from a long straight seawall; let $O$ be the point of the wall closest to it. The lamp rotates at a constant rate, one full turn every $12$ seconds, and at $t=0$ s its beam points straight at $O$. Let $p(t)$ be the distance, in metres, from $O$ to the lit spot, measured positively in the direction the beam sweeps.

   1. Show that $p(t)=200\tan \left(\frac{\pi }{6}t\right)$.
   2. Find $p(1)$, to the nearest hundredth of a metre.
   3. Find the first time the spot is $200\sqrt{3}$ m from $O$.
   4. What happens at $t=3$ s?
8. Q8Solving Problems Involving the Cosine Function

   At a wharf, the depth of water in metres is modelled by $d(t)=1.8\cos (\frac{\pi }{6}(t-4))+5.2$, where $t$ is the number of hours after midnight.

   1. What is the depth at 07:00?
   2. At what times between midnight and noon is the depth exactly $6.1$ m?
9. Q9Solving Problems Involving the Sine Function

   The height, in metres, of the tip of a wind-turbine blade above the ground is $H(t)=32\sin (\frac{\pi }{5}(t-2.5))+45$, where $t$ is in seconds.

   1. How long does one full rotation take, and how high is the tip at its highest?
   2. Find the first time $t\ge 0$ at which the tip is $61$ m above the ground.
10. Q10The Cosine Function

    Let $f(x)=a\cos (b(x-h))+k$ with $a\ne 0$ and $b\ne 0$. For each claim below, state whether it is always, sometimes or never true, and justify your choice in one or two lines (with a counterexample when the answer is “sometimes” or “never”).

    1. The range of $f$ is $[k-a,k+a]$.
    2. $f(h)$ is the maximum value of $f$.
    3. $f$ has exactly three zeros.
    4. The zeros of $f$ are spaced half a period apart.
11. Q11The Inverse of the Cosine Function (Arccos)
    1. Evaluate, in radians: $\arccos (-\frac{\sqrt{3}}{2})$, $\arccos (0)$ and $\arccos (1)$.
    2. State the domain and the range of the function $\arccos$.
    3. Explain why $\arccos \left(\cos \frac{7\pi }{4}\right)\ne \frac{7\pi }{4}$, and give its exact value.
12. Q12The Inverse of the Sine Function (Arcsin)
    1. Evaluate, in radians: $\arcsin \left(\frac{1}{2}\right)$, $\arcsin (-\frac{\sqrt{2}}{2})$ and $\arcsin (-1)$.
    2. State the domain and the range of $\arcsin$, using Québec interval notation.
    3. Using the value of $\arcsin (-\frac{1}{2})$, find every $x\in [0,2\pi ]$ such that $\sin x=-\frac{1}{2}$.
13. Q13The Inverse of the Tangent Function (arctan)
    1. Evaluate, in radians: $\arctan (1)$, $\arctan (-\sqrt{3})$ and $\arctan (0)$.
    2. State the domain and the range of $\arctan$, using Québec interval notation, and say why the range brackets are open.
    3. A loading ramp rises $1.2$ m over a horizontal run of $8$ m. Find the angle it makes with the horizontal, in radians to the nearest hundredth.
14. Q14The Law of Cosines

    A surveyor walks two sides of a triangular lot: $48$ m, then $65$ m, with an angle of ${112}^{\circ }$ between them.

    1. Find the length of the third side, to the nearest tenth of a metre.
    2. Find the smallest angle of the lot, to the nearest tenth of a degree.
15. Q15The Law of Sines

    Two fire-lookout towers $A$ and $B$ stand on a straight ridge, $3.2$ km apart. Both spot the same smoke plume $S$ on the far side of the valley. From tower $A$, the sight line to $S$ makes an angle of ${47}^{\circ }$ with the ridge $AB$; from tower $B$, the sight line makes an angle of ${68}^{\circ }$ with the ridge $BA$. How far is the plume from tower $A$? Round to the nearest hundredth of a kilometre.

16. Q16The Properties of the Cosine Function (Sine)

    Let $f(x)=3\cos (2(x-\frac{\pi }{4}))-1$.

    1. State the amplitude, the period and the range of $f$.
    2. Give the coordinates of all maxima and of all minima of $f$.
    3. Compute $f(0)$.
    4. How many maxima does $f$ have on the interval $[0,4\pi ]$?
17. Q17The Properties of the Sine Function (Cosine)

    Let $g(x)=-4\sin (\frac{\pi }{2}(x+1))+2$.

    1. Rewrite the rule in the form $a\sin (b(x-h))+k$ and state $a$, $b$, $h$, $k$.
    2. Give the period and the range of $g$.
    3. Give the coordinates of one maximum and one minimum.
    4. On which intervals is $g$ increasing?
18. Q18The Properties of the Tangent Function

    Let $f(x)=2\tan (\frac{1}{2}(x-\frac{\pi }{3}))$.

    1. Give the period of $f$ and the equations of all its asymptotes.
    2. Give the domain and the range of $f$.
    3. Find all the zeroes of $f$.
    4. Evaluate $f(\frac{\pi }{3}+\frac{\pi }{2})$ exactly.
19. Q19The Role of Parameters in a Cosine Function

    Consider $f(x)=4\cos (2(x-\frac{\pi }{4}))-1$. Each change below alters exactly one parameter. For each, state the geometric effect on the graph and say whether the period and the range change.

    1. $a$ is replaced by $-4$.
    2. $b$ is replaced by $4$.
    3. $k$ is replaced by $5$.
    4. $h$ is replaced by $\frac{\pi }{4}+\pi$.
20. Q20The Role of Parameters in a Tangent Function

    Four tangent functions are given: $$f_1(x)=3\tan (2x),f_2(x)=-5\tan (2x),f_3(x)=3\tan (2(x-\frac{\pi }{4})),f_4(x)=3\tan \left(\frac{1}{2}x\right)+3.$$

    1. Sort the four functions into groups having the same period, and give each period.
    2. Exactly two of them have the same set of asymptotes. Which two, and what are those asymptotes?
    3. Explain why the parameters $a$ and $k$ can never move the asymptotes of a tangent function.
21. Q21The Role of the Parameters in a Sine Function

    While revising, Simon writes: “In $f(x)=a\sin (b(x-h))+k$, making $|b|$ larger makes the graph both taller and faster.” Only half of his sentence is right. State precisely what $b$ does and does not control, back your answer up with a specific pair of rules and their ranges and periods, and say what a negative value of $b$ does to the graph.

22. Q22The Sine Function

    Answer using the unit circle, without a calculator.

    1. State the domain, the range and the period of the basic function $f(x)=\sin x$.
    2. Explain why $\sin (\pi -\theta )=\sin \theta$ for every real number $\theta$.
    3. Use b) to give the exact value of $\sin \frac{2\pi }{3}$.
    4. Give all $x\in [0,2\pi ]$ for which $\sin x=-\frac{\sqrt{2}}{2}$.
23. Q23The Tangent Function

    Answer the following about the basic function $f(x)=\tan x$.

    1. Using $\tan x={\displaystyle \frac{\sin x}{\cos x}}$, explain why $f$ is undefined exactly at $x=\frac{\pi }{2}+\pi n$, $n\in \mathbb{Z}$.
    2. State the domain, the range and the period of $f$.
    3. Explain why it makes no sense to speak of the amplitude of $f$.
    4. Give the exact value of $\tan \frac{5\pi }{3}$.
24. Q24The Unit Circle

    Work from the unit circle; give exact values.

    1. Give the coordinates of the point associated with a rotation of $\frac{11\pi }{6}$.
    2. Evaluate $\cos \frac{7\pi }{6}$, $\sin \frac{5\pi }{3}$ and $\tan \frac{3\pi }{4}$.
    3. Which rotation in $[0,2\pi [$ lands on the point $(-\frac{1}{2},\frac{\sqrt{3}}{2})$?
25. Q25The Zeroes of a Trigonometric Function

    Find all the zeroes of $f(x)=2\cos (2(x-\frac{\pi }{6}))+1$, then list those that belong to $[0,2\pi ]$.

26. Q26Trigonometric Angles (Radians)
    1. Convert ${135}^{\circ }$ and ${300}^{\circ }$ to radians, in exact form.
    2. Convert $\frac{7\pi }{12}$ rad to degrees.
    3. A rotating sprinkler is set to sweep an angle of $\frac{5\pi }{6}$ rad and throws water $12$ m. Find the exact length of the arc traced by the outer edge of the spray and the exact area watered, then give each to the nearest hundredth.
27. Q27Trigonometric Functions

    In a research greenhouse, the air temperature $t$ hours after midnight is modelled by $$T(t)=3.5\sin (\frac{\pi }{12}(t-8))+21,0\le t\le 24,$$ where $T$ is in degrees Celsius.

    1. What is the period of the model, and what does it mean here?
    2. What is the warmest temperature reached, and at what time of day?
    3. Compute $T(2)$ and interpret the value.
28. Q28Trigonometry and Metric Relations

    This topic supplies several distinct tools. For each situation below, name the one tool you would reach for first and give a one-sentence reason. Choose from: the law of sines, the law of cosines, the unit circle, an inverse trigonometric function (arcsin, arccos, arctan), a sinusoidal model $a\sin (b(x-h))+k$.

    1. In a triangle you know two sides and the angle between them, and you want the third side.
    2. In a triangle you know two angles and the side opposite one of them, and you want another side.
    3. You need the exact value of $\cos \frac{5\pi }{3}$, with no calculator.
    4. The height of water in a lock rises and falls, repeating every $9$ minutes.
    5. You know the two legs of a right triangle and want its acute angles.

The 23 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The unit circle is a memory device, not a table to memorise (Q22, Q23, Q24)

A rotation of θ from the positive horizontal axis lands on the point (cos θ, sin θ). That single sentence generates everything: the cosine is the abscissa, the sine is the ordinate, and the tangent is their quotient, which is why it is undefined exactly where the abscissa is zero.

The symmetries asked for in Q22 and Q24 come from the same picture. The points for θ and π − θ are mirror images in the vertical axis, so they have opposite abscissas and the same ordinate — which is exactly sin(π − θ) = sin θ. Adding π sends a point to the diametrically opposite one, so both coordinates change sign. Replacing θ by −θ reflects in the horizontal axis, so the abscissa survives and the ordinate flips. Learning the picture is faster and more reliable than learning the four identities.

Radians, arc length and sector area (Q26)

Multiply by π/180 to go from degrees to radians and by 180/π to come back — and if you can never remember which, note that a radian is much bigger than a degree, so a number of degrees must shrink.

Both formulas are true only with θ in radians — that is the entire reason radians exist, and it is what makes them the natural unit for the rest of this sheet. The sprinkler question also carries a units trap of its own: the radius and the arc must be measured in the same unit before you multiply, and the answer is asked for exactly and then rounded, so keep the π until the final line.

Reading a, b, h, k off a rule — after factoring b out (Q4, Q5, Q16, Q17, Q19, Q21)

In the standard form a·sin(b(x − h)) + k, and identically for cosine and tangent:

|a| is the amplitude — half the distance between the maximum and the minimum, which is why a tangent function has no amplitude at all. k is the midline y = k, and the range is [k − |a|, k + |a|]. b controls only the horizontal behaviour: the period is 2π/|b| for sine and cosine and π/|b| for tangent. h is the horizontal shift.

The sign of a decides where the curve starts. With a > 0 a cosine has a maximum at x = h; with a < 0 it has a minimum there, and every extremum swaps. For a sine with a > 0, x = h is a rising midline crossing; with a < 0 it is a falling one. Q16 and Q17 ask for the coordinates of all maxima and minima, which means a general expression with an integer n in it — spaced one full period apart, with the minima half a period from the maxima.

Finding the rule from a description (Q1, Q2, Q3)

Three questions, one procedure. Whatever you are told, a and k come from the extremes and b comes from the period:

The midline is the average of the two extremes; the amplitude is half their difference. What changes between the questions is how the period is disguised, and it is worth knowing the three disguises by heart. A maximum and the next minimum are half a period apart (Q1). A rising midline crossing and the next maximum are a quarter period apart (Q2). Two consecutive asymptotes of a tangent function are a whole period apart — and for tangent the period is π/b, not 2π/b (Q3).

Then h is placed by whichever landmark the question hands you: the abscissa of a maximum for a cosine with a > 0, the abscissa of a rising midline crossing for a sine with a > 0, the centre of a branch for a tangent. Q3 gives a further point on the curve, which is how a is found once b, h and k are known: substitute the point and solve the resulting one-unknown equation.

Tangent behaves differently, and the differences are the marks (Q6, Q18, Q20, Q23)

Its period is π/|b|, half of what the same b would give a sine. Its asymptotes are where the argument hits π/2 + πn, so solving b(x − h) = π/2 + πn for x gives the whole family at once — and the answer is a family, with an integer n in it, not a single line. Its zeroes are where the argument is πn. The centre of each branch sits at a zero, exactly midway between two consecutive asymptotes, and the branch has point symmetry about that centre, which is what lets Q3 predict a second point from a first.

Two things it does not have: an amplitude and a bounded range. On each branch a tangent function takes every real value, so it has no maximum and no minimum, and half the distance between them is not a quantity that exists. Its range is all the reals — with the domain excluding the asymptotes. And a and k can never move an asymptote (Q20): an asymptote is where the function is undefined, which depends only on the argument b(x − h). Multiplying an undefined value by a or adding k to it cannot make it defined. Finally, the branch is increasing exactly when a and b have the same sign — two reversals cancel.

Solving a trigonometric equation: one solution is never the answer (Q8, Q9, Q12, Q25)

Q25 asks for all the zeroes and then the ones inside an interval, which is this procedure with the right-hand side placed by the constant term. An inequality — Q8's "at what times is the depth exactly this", or a modelling question asking when a quantity stays above a threshold — is the same work with one extra step: find the two boundary angles, then keep the stretch of the cycle between them, and read off an interval of time rather than a list of instants.

The inverse functions and why their ranges are restricted (Q11, Q12, Q13)

A sine takes the same value infinitely often, so it has no inverse until its domain is cut down to a stretch on which it is one-to-one. That is what the ranges of the inverse functions are: arccos returns an angle in [0, π], arcsin in [−π/2, π/2], and arctan in ]−π/2, π/2[ — with reversed brackets, because the tangent is undefined at those two ends and no input can ever be sent there. Their domains are [−1, 1] for arcsin and arccos, since a sine or cosine never leaves that interval, and all the reals for arctan, since a tangent takes every real value on one branch.

Modelling: tides, blades and beams (Q7, Q8, Q9, Q27)

A quantity that oscillates between a fixed maximum and a fixed minimum with a constant period is exactly what a sinusoidal rule describes, and the modelling questions are the graphing skills read in context. The vocabulary translates directly: the midline is the average level, the amplitude is how far it swings either side, the period is how long one full cycle takes, and h is anchored by whichever event the problem times for you — a high tide, a peak temperature, the moment the rider boards.

A rotating beam is the exception that needs a tangent instead: the lit spot's distance along a straight wall is the perpendicular distance times the tangent of the swept angle, which is why the spot accelerates away and why the beam going parallel to the wall is an asymptote, not a very large number. Q7 asks what happens at that instant, and the honest answer is that there is no lit spot at all.

Two habits for these questions. Convert the answer back into the language of the problem — "at 06:00", "1.56 m below mean sea level" — because the interpretation carries its own mark. And check the model against a value you were given before using it for anything else.

Always, sometimes, never (Q10)

Q10 gives four claims about a cosine function and asks for a verdict with a justification. "Sometimes" and "never" both need an explicit counterexample — a specific rule, with the arithmetic shown — while "always" needs a general argument. Two of the claims turn on the sign of a, which is the recurring theme of this whole topic. A third is about zeroes: a periodic function that has one zero automatically has infinitely many, one per period, so "exactly three zeroes" is impossible rather than unusual — the count is either none or infinite. Recognising an impossible claim, rather than an unlikely one, is what the question is testing.

Law of sines or law of cosines? (Q14, Q15, Q28)

Both relate the sides of a triangle to its angles, and the choice is made by what you were given, not by what you want.

Two checks worth building in. The largest angle always faces the longest side and the smallest faces the shortest — so if your smallest angle is opposite the longest side, something is wrong before you go any further. And the three angles must sum to 180°, which costs nothing to verify.

Q28 asks you to name the right tool for five situations without solving anything. It looks like the easiest question on the sheet and it is the one that predicts an exam result best, because on an exam nobody tells you which tool the problem wants.

Preview all 15 pages

Click any page to open the full PDF.

Getting the most out of it

Check the calculator mode before the first question

Everything here is in radians, and a calculator in degree mode fails silently — it returns a number, just the wrong one. Test it with sin(π/6), which must come out as exactly 0.5, and test it again whenever the calculator has left your hands.

Sketch the curve even for the algebra questions

Midline, amplitude band, one period marked in quarters. It takes twenty seconds and it turns the equation questions into counting exercises: you can see how many times the curve crosses the level you are solving for, so you know before you start whether to expect two solutions per period or four in the interval.

Do the tool-naming question last, as a review

Come back to the question that asks which tool fits which situation after you have worked through the rest. By then it stops being vocabulary and becomes a summary of everything on the sheet — which is exactly the form an exam problem takes, since an exam never tells you whether it wants the law of sines or a sinusoidal model.

Common questions