MHF4U · Unit 4 test review

Trigonometric Functions: Test Review

Everything for the Unit 4 test on one page: the key facts from all 5 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Radian Measure

One radian is the central angle subtending an arc equal in length to the radius: $θ = \frac{a}{r}$ .
One full revolution is $2 π$ radians, so $π rad = 180°$ and $1 rad \approx 57.3°$ .
Degrees to radians: multiply by $\frac{π}{180}$ . Radians to degrees: multiply by $\frac{180}{π}$ .
Arc length $a = r θ$ and sector area $A = \frac{1}{2} r^{2} θ$ , both with $θ$ in radians.
Angular velocity $ω = \frac{θ}{t}$ (rad/s); linear speed at radius $r$ is $v = r ω$ ; and $1$ rev/min $= 2 π$ rad/min.
Radians are unitless (a length over a length), so an angle written as a bare number is in radians.

The Unit Circle and Exact Values

On the unit circle, the point at angle $θ$ is $(cos θ, sin θ)$ , and $tan θ = \frac{sin θ}{cos θ}$ .
Because $x^{2} + y^{2} = 1$ on the circle, $sin^{2} θ + cos^{2} θ = 1$ for every angle.
Related angles: $π - θ$ in QII, $θ - π$ in QIII, $2 π - θ$ in QIV.
CAST: all positive in QI, only sine in QII, only tangent in QIII, only cosine in QIV.
Coterminal angles differ by a multiple of $2 π$ and share identical trig values.
$sin \frac{π}{6} = \frac{1}{2}$ , $cos \frac{π}{6} = \frac{3}{2}$ , $sin \frac{π}{4} = cos \frac{π}{4} = \frac{2}{2}$ , $sin \frac{π}{3} = \frac{3}{2}$ , $tan \frac{π}{4} = 1$ .

Trigonometric Graphs and Transformations

$y = sin x$ and $y = cos x$ : period $2 π$ , amplitude $1$ , axis $y = 0$ , range $[- 1, 1]$ , domain all real numbers.
$y = tan x$ : period $π$ , range all real numbers, vertical asymptotes at $x = \frac{π}{2} + nπ$ , zeros at $x = nπ$ .
In $y = a sin (k (x - d)) + c$ : amplitude $= ∣ a ∣$ , period $= \frac{2 π}{∣ k ∣}$ , phase shift $= d$ (right if positive), axis $= y = c$ .
Maximum $= c + ∣ a ∣$ and minimum $= c - ∣ a ∣$ .
From a graph: $c = \frac{max + min}{2}$ , $a = \frac{max - min}{2}$ , $k = \frac{2 π}{period}$ .
Always factor $k$ out of the bracket before reading the phase shift.

Modelling with Sinusoidal Functions

Axis $c = \frac{max + min}{2}$ ; amplitude $∣ a ∣ = \frac{max - min}{2}$ .
$k = \frac{2 π}{period}$ , with the period in the model's time units.
Start at a maximum: $+ cos$ ; at a minimum: $- cos$ ; on the axis and rising: $+ sin$ . Otherwise set $d$ to the time of a maximum and use $+ cos$ .
To predict, substitute in radian mode; to find when a value occurs, isolate the trig function, take the inverse, find the second angle in the cycle, convert to time, then add periods.
For cosine the second angle is $2 π - θ$ ; for sine it is $π - θ$ .
Interpret every parameter with units: amplitude as radius or half-range, axis as average level, period as cycle length, d as the time of a key point.

Reciprocal Trigonometric Functions

$csc θ = \frac{1}{sin θ}$ , $sec θ = \frac{1}{cos θ}$ , $cot θ = \frac{1}{tan θ} = \frac{cos θ}{sin θ}$ .
A reciprocal graph has a vertical asymptote wherever its base function equals zero.
Where the base function peaks at $\pm 1$ , the reciprocal touches it and turns — a local minimum above the axis, a local maximum below it.
$y = csc x$ and $y = sec x$ have period $2 π$ and range $y \leq - 1$ or $y \geq 1$ ; nothing lands strictly between $- 1$ and $1$ .
$y = cot x$ has period $π$ , range all real numbers, asymptotes at $x = nπ$ , and zeros at $x = \frac{π}{2} + nπ$ .
Reciprocals keep the sign of the base function, so CAST applies to $csc$ , $sec$ , and $cot$ unchanged.

Where marks get lost

Leaving the calculator in DEG mode. Every trig value in this unit comes out wrong and, worse, it comes out looking plausible — $sin (π /6)$ in degree mode returns about $0.0091$ instead of $0.5$ . (Radian Measure)
Using $a = r θ$ or $A = \frac{1}{2} r^{2} θ$ with the angle still in degrees. Convert to radians first, every single time. (Radian Measure)
Multiplying by $\frac{180}{π}$ when converting degrees to radians (the factor is upside down). Check the size of your answer: $90°$ must become about $1.57$ , not about $5157$ . (Radian Measure)
Reporting radian answers as decimals when an exact value is expected — write $\frac{3 π}{4}$ , not $2.36$ , unless the question explicitly asks for a decimal. (Radian Measure)
Forgetting that revolutions per minute must be multiplied by $2 π$ (not by $360$ ) before they are radians per minute. (Radian Measure)
Swapping the coordinates: writing the unit-circle point as $(sin θ, cos θ)$ . Cosine comes first because cosine is the horizontal ( $x$ ) coordinate. (The Unit Circle and Exact Values)
Finding the related angle correctly and then forgetting the sign, so $cos \frac{5 π}{6}$ comes out as $+ \frac{3}{2}$ instead of $- \frac{3}{2}$ . (The Unit Circle and Exact Values)
Using the wrong related-angle formula for the quadrant — applying $π - θ$ in quadrant III instead of $θ - π$ produces a negative related angle, which is an immediate signal that you picked the wrong row. (The Unit Circle and Exact Values)
Assuming a negative angle automatically gives a negative answer. The sign depends only on which quadrant the arm lands in: $cos (- \frac{π}{3}) = + \frac{1}{2}$ , because $- \frac{π}{3}$ is in quadrant IV where cosine is positive. (The Unit Circle and Exact Values)
Giving only one solution when solving an equation over $[0, 2 π)$ . Except for the values 1 and −1, every sine or cosine value between −1 and 1 occurs at exactly two angles in $[0, 2 π)$ . (The Unit Circle and Exact Values)
Using $\frac{360°}{∣ k ∣}$ for the period out of habit. In radians the period of sine and cosine is $\frac{2 π}{∣ k ∣}$ . (Trigonometric Graphs and Transformations)
Reading the phase shift straight out of an unfactored bracket — $y = sin (2 x - π /3)$ shifts by $\frac{π}{6}$ , not $\frac{π}{3}$ , because $k$ must be factored out first. (Trigonometric Graphs and Transformations)
Giving tangent a period of $2 π$ or an amplitude. Tangent repeats every $π$ , and since its range is all real numbers it has no maximum, no minimum, and no amplitude. (Trigonometric Graphs and Transformations)
Reading the sign of $d$ backwards: $(x - d)$ with a positive $d$ shifts RIGHT, so $y = cos (x + π /4)$ means $d = - \frac{π}{4}$ , a shift LEFT. (Trigonometric Graphs and Transformations)
Confusing amplitude with the full vertical spread. Amplitude is HALF the distance from minimum to maximum, so a curve running from $- 1$ to $7$ has amplitude $4$ , not $8$ . (Trigonometric Graphs and Transformations)
Using the period as $k$ : a 12.4 h tide needs $k = \frac{2 π}{12.4} \approx 0.507$ , not $12.4$ . (Modelling with Sinusoidal Functions)
Evaluating the model with the calculator in degree mode, which gives wrong values with no warning. (Modelling with Sinusoidal Functions)
Reporting only the calculator's first inverse-cosine answer and missing the second time in the cycle. (Modelling with Sinusoidal Functions)
Forgetting to undo the horizontal shift: after dividing by $k$ , you must still add $d$ to get the actual time. (Modelling with Sinusoidal Functions)
Using +cos for a Ferris wheel when the rider boards at the bottom, which puts the rider at the top at t = 0. (Modelling with Sinusoidal Functions)
Pairing the names by first letter: assuming cosecant inverts cosine. It does not — cosecant inverts SINE and secant inverts COSINE. (Reciprocal Trigonometric Functions)
Confusing $csc θ$ with $sin^{- 1} θ$ . The first is $\frac{1}{sin θ}$ , a number; the second is an angle. The exponent $- 1$ on a trig name always means the inverse function. (Reciprocal Trigonometric Functions)
Putting the asymptotes of $y = cot x$ where tangent's are. Cotangent is $\frac{cos x}{sin x}$ , so its asymptotes are at $x = nπ$ where SINE is zero, and its zeros sit where cosine is zero. (Reciprocal Trigonometric Functions)
Giving $y = csc x$ a period of $π$ because it looks like repeating arches. Consecutive arches point in opposite directions, so one full repeat takes $2 π$ . (Reciprocal Trigonometric Functions)
Leaving an answer as $\frac{2}{3}$ when an exact value is requested. Rationalize the denominator to $\frac{2 3}{3}$ . (Reciprocal Trigonometric Functions)

Practice test

25 questions from across the unit. Each one shows the lesson it comes from.

Radian Measure
1
Convert $135°$ to radians. Give the decimal value, rounded to 2 decimals.
Radian Measure
2
Why does one full revolution measure exactly $2 π$ radians?
Radian Measure
3
A sector of a circle of radius $4$ m has an arc length of $6$ m. Find the central angle in radians.
Radian Measure
4
Arrange these angles from smallest to largest.
1. 1 $150°$
2. 2 $2$ radians
3. 3 $\frac{π}{3}$
4. 4 $\frac{π}{6}$
5. 5 $45°$
Radian Measure
5
In the formula $a = r θ$ , what unit must $θ$ be expressed in?
The Unit Circle and Exact Values
6
Evaluate $4 cos \frac{π}{3}$ exactly.
The Unit Circle and Exact Values
7
Evaluate $tan \frac{4 π}{3}$ . Give the decimal value, rounded to 3 decimals.
The Unit Circle and Exact Values
8
In which quadrant is $sin θ$ negative while $tan θ$ is positive?
The Unit Circle and Exact Values
9
Evaluate $sin \frac{11 π}{6}$ exactly (enter it as a decimal).
The Unit Circle and Exact Values
10
Solve $cos θ = - \frac{2}{2}$ on $0 \leq θ < 2 π$ and give the SMALLER of the two solutions, as a decimal rounded to 3 decimals.
Trigonometric Graphs and Transformations
11
Find the period of $y = sin (3 x)$ . Give the decimal value, rounded to 2 decimals.
Trigonometric Graphs and Transformations
12
What is the equation of the axis of $y = - 6 cos (2 (x - \frac{π}{3})) + 1$ ?
Trigonometric Graphs and Transformations
13
For $y = 5 sin (2 (x - \frac{π}{4})) + 3$ , find $y$ when $x = \frac{π}{2}$ .
Trigonometric Graphs and Transformations
14
Why does $y = tan x$ have vertical asymptotes at $x = \frac{π}{2} + nπ$ ?
Trigonometric Graphs and Transformations
15
A sinusoid has a maximum of $11$ , a minimum of $3$ , and a period of $π$ . Find the value of $k$ .
Modelling with Sinusoidal Functions
16
A tide has a period of $12.4$ h. Find $k$ for a model in radians, to 3 decimal places.
Modelling with Sinusoidal Functions
17
Monthly average temperatures range from $- 5^{\circ}$ C to $2 2^{\circ}$ C. What is the axis $c$ of the model, in degrees Celsius?
°C
Modelling with Sinusoidal Functions
18
For $h (t) = - 15 cos (π t) + 17$ , when does the rider first reach $25$ m? Answer in minutes to 2 decimal places.
min
Modelling with Sinusoidal Functions
19
Put the steps for finding when a sinusoidal model reaches a value in order.
1. 1Divide by k and add d to convert angles to times
2. 2Find the second angle in the same cycle
3. 3Add whole periods for later occurrences
4. 4Set the model equal to the target value
5. 5Isolate the sine or cosine
6. 6Take the inverse to get the first angle
Modelling with Sinusoidal Functions
20
Using $D (t) = 3.25 sin (\frac{2 π}{365} (t - 80)) + 12.15,$ predict the day length on day $100$ , in hours to 1 decimal place.
h
Reciprocal Trigonometric Functions
21
Evaluate $sec \frac{π}{3}$ exactly.
Reciprocal Trigonometric Functions
22
Where does $y = sec x$ have vertical asymptotes?
Reciprocal Trigonometric Functions
23
Which is the range of $y = csc x$ ?
Reciprocal Trigonometric Functions
24
Arrange these exact values from smallest to largest.
1. 1 $cot \frac{π}{4}$
2. 2 $sec \frac{π}{3}$
3. 3 $csc \frac{π}{4}$
4. 4 $cot \frac{π}{2}$
Reciprocal Trigonometric Functions
25
Is $csc θ$ the same thing as $sin^{- 1} θ$ ? Answer yes or no.

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…