OMPT drill — Trigonometry

Trigonometric functions

Sine, cosine, tangent as functions: graphs, amplitude, period, and exact values. Learn the unit circle once, properly, and half this topic is free. Keep a unit circle sketch next to you for the first pass, then redo the set without it.

Tested inOMPT-B OMPT-D OMPT-G

Lesson

Put a point on the unit circle at angle $θ$ , measured anticlockwise from the positive $x$ -axis. Its coordinates are the whole subject: the $x$ -coordinate is $cos θ$ , the $y$ -coordinate is $sin θ$ , and the ratio $\frac{s i n θ}{c o s θ}$ is $tan θ$ . Every exact value, every sign rule, every identity you will meet is a statement about that one moving point. Learn the picture and you get the facts for free; learn the facts without the picture and they leak.

Two consequences fall out immediately. Since the point lives on a circle of radius one, sine and cosine are trapped between $- 1$ and $1$ forever. And since adding $2 π$ to the angle walks you back to the same point, both functions repeat with period $2 π$ , which is what makes their graphs waves. The exam earns its questions from a small table of exact values at the landmark angles, plus reading amplitude and period off transformed waves like $3 sin (2 x)$ .

Problem 1Multiple choice

What is the period of $f (x) = 3 sin (4 x - π) + 2$ ?

A $2 π$
B $π$
C $\frac{π}{2}$
D $\frac{π}{4}$

Show the worked solution

Answer: C — $\frac{π}{2}$

Only the coefficient of $x$ inside the sine affects the period. With $b = 4$ , the period is $\frac{2 π}{b} = \frac{2 π}{4} = \frac{π}{2}$ . The amplitude $3$ , phase shift, and vertical shift $2$ play no role here.

Problem 2Numeric answer

Evaluate $cos (\frac{5 π}{6})$ as a decimal, rounded to two decimals.

Show the worked solution

Answer: -0.87

The angle $\frac{5 π}{6}$ sits in the second quadrant with reference angle $\frac{π}{6}$ . Cosine is negative there, so $cos \frac{5 π}{6} = - cos \frac{π}{6} = - \frac{3}{2} \approx - 0.87$ .

Problem 3Multiple choice

Which transformation of $y = cos x$ produces the graph of $y = - 2 cos (x - \frac{π}{3})$ ?

AStretch vertically by 2, shift left $\frac{π}{3}$
BStretch vertically by 2, reflect in the x-axis, shift right $\frac{π}{3}$
CCompress vertically by 2, shift right $\frac{π}{3}$
DReflect in the y-axis, shift right $\frac{π}{3}$

Show the worked solution

Answer: B — Stretch vertically by 2, reflect in the x-axis, shift right $\frac{π}{3}$

The factor $- 2$ does two things: the $2$ stretches the graph vertically and the minus sign flips it across the x-axis. Inside the argument, $x - \frac{π}{3}$ moves the graph right by $\frac{π}{3}$ — the inside shift always works opposite to its sign.

Problem 4Numeric answer

The water depth at a jetty is modelled by $d (t) = 5 + 3 sin (\frac{π t}{6})$ metres, with $t$ in hours after midnight. What is the depth at $t = 9$ ?

Show the worked solution

Answer: 2

I substitute directly: $\frac{π \cdot 9}{6} = \frac{3 π}{2}$ , and $sin \frac{3 π}{2} = - 1$ . So $d (9) = 5 + 3 (- 1) = 2$ metres — this moment happens to be dead low tide.

Problem 5Spot the error

A student finds the range of $y = 4 - 2 sin x$ . Step 1: $sin x$ takes values in $[- 1, 1]$ . Step 2: so $- 2 sin x$ takes values in $[- 2, 2]$ . Step 3: adding 4 to the endpoints $- 2$ and $1$ gives the range $[2, 5]$ . Which step contains the error?

Show the worked solution

Answer: Step 3

Steps 1 and 2 are fine. In Step 3 the student added 4 to $- 2$ and then slipped back to the original $1$ from Step 1 instead of the $2$ from Step 2. Adding 4 to the whole interval $[- 2, 2]$ gives $[2, 6]$ , so the range is $[2, 6]$ .

Problem 6Numeric answer

Given $tan θ = - \frac{3}{4}$ with $θ$ in the second quadrant, find $sin θ$ as a decimal.

Show the worked solution

Answer: 0.6

A tangent of $\frac{3}{4}$ in size suggests a 3-4-5 triangle, so the hypotenuse is 5. In quadrant II sine is positive and cosine negative, which is exactly what makes the tangent negative. Therefore $sin θ = \frac{3}{5} = 0.6$ (and $cos θ = - \frac{4}{5}$ , consistent with $tan θ = - \frac{3}{4}$ ).