Year 11 Maths Advanced Module 2 ⏱ ~35 min Lesson 15 of 15

Review and Connections

You have reached the final lesson of Module 2. This lesson brings together everything you have learned about trigonometric functions and graphs — from exact values and identities to sketching, solving, and modelling. Use this review to solidify your understanding before tackling the Module Quiz.

🔗

Printable worksheet

Download this lesson's worksheet

Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.

Download PDF Open printable version

Think First

Without looking back at your notes, try to list as many connections as you can between the ideas in this module. For example: how are the Pythagorean identities connected to the unit circle? How are phase shifts connected to horizontal translations? How is solving trig equations connected to graph intersections?

Type your initial response below — you will revisit this at the end of the lesson.

Write your initial response in your book. You will revisit it at the end of the lesson.

Write your initial thinking in your book

Saved

📖 Know

Key Facts

All six trig functions and their reciprocal relationships
The three Pythagorean identities
The domains, ranges, periods, and key features of all trig graphs

💡 Understand

Concepts

How all trig ideas connect back to the unit circle
How transformations affect trig graphs
How trig functions model periodic real-world phenomena

✅ Can Do

Skills

Solve problems combining identities, exact values, and equations
Analyse and sketch transformed trig graphs
Build and use trig models for applications

Misconceptions to Fix

✗

Wrong: √(a + b) = √a + √b.

✗

Right: The square root of a sum is not the sum of square roots; √(a+b) cannot be simplified this way.

📚 Core Content

Key Terms

Periodic FunctionA function that repeats its values at regular intervals.

RadianThe natural unit for angle measure in calculus and physics.

AmplitudeThe maximum displacement from the centre line of a periodic function.

Phase ShiftA horizontal translation of a trigonometric graph.

IdentityAn equation that is true for all valid values of the variable.

Exact ValueA trig value expressed using surds and fractions, not decimals.

🔄

Module 2 at a Glance

Inquiry Question 1: Trigonometry

Exact values of sine, cosine, and tangent at special angles ($0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$ and multiples)
Reciprocal trig functions: $\csc x$, $\sec x$, $\cot x$ and their relationships to sine, cosine, and tangent
Pythagorean identities: $\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$, $1 + \cot^2 x = \csc^2 x$
Complementary angle identities (co-functions)

Inquiry Question 2: Trigonometric Functions and Graphs

Domains and ranges of all six trig functions
Graphs of sine and cosine: amplitude, period, intercepts, maxima, minima
Graphs of tangent and cotangent: asymptotes, period $\pi$, branches
Transformations: $y = a\sin(b(x - c)) + d$ and the meaning of each parameter
Phase shifts as horizontal translations
Solving equations graphically by finding intersections
Modelling periodic phenomena with trig functions

The unit circle: the heart of it all. Every concept in this module — exact values, identities, signs in different quadrants, the shapes of graphs, domain restrictions — can be traced back to the unit circle. If you ever feel lost, return to the unit circle. It is the single most powerful diagram in trigonometry.

🔗

Making Connections

From Identities to Graphs

The identity $\sin^2 x + \cos^2 x = 1$ is why the graphs of $y = \sin x$ and $y = \cos x$ are both bounded between $-1$ and $1$. The Pythagorean identity is the algebraic expression of the geometric fact that points on the unit circle satisfy $x^2 + y^2 = 1$.

From Equations to Graphs

Solving $\sin x = k$ is equivalent to finding the $x$-coordinates where the horizontal line $y = k$ intersects the sine curve. This graphical viewpoint explains why there can be zero, one, two, or infinitely many solutions depending on the value of $k$ and the domain.

From Graphs to Models

Once you understand how amplitude, period, phase shift, and vertical shift transform the basic sine and cosine graphs, you can reverse the process: take real-world data, identify these features, and write a mathematical model that predicts future behaviour.

🧮 Worked Examples

Worked Example 1 — Combined Identity and Exact Value

Stepwise

If $\sin \theta = \frac{1}{3}$ and $\frac{\pi}{2} < \theta < \pi$, find the exact value of $\tan \theta$.

1

Find $\cos \theta$ using the Pythagorean identity

\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{1}{9} = \frac{8}{9}
2

Determine the sign

In QII, $\cos \theta < 0$, so $\cos \theta = -\frac{\sqrt{8}}{3} = -\frac{2\sqrt{2}}{3}$.
3

Calculate $\tan \theta$

\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{3}}{-\frac{2\sqrt{2}}{3}} = -\frac{1}{2\sqrt{2}} = -\frac{\sqrt{2}}{4}

✓ Answer $-\frac{\sqrt{2}}{4}$

Worked Example 2 — Sketching a Transformed Graph

Stepwise

Sketch $y = 2\sin\left(x + \frac{\pi}{3}\right) - 1$ for $-\frac{\pi}{3} \leq x \leq \frac{5\pi}{3}$ and state the range.

1

Identify the transformations

Amplitude = 2, Period = $2\pi$, Phase shift = $\frac{\pi}{3}$ left, Vertical shift = 1 down.
2

Find key points

Starts at midline going up when $x = -\frac{\pi}{3}$ ($y = -1$)
Max at $x = \frac{\pi}{6}$ ($y = 1$)
Midline at $x = \frac{2\pi}{3}$ ($y = -1$)
Min at $x = \frac{7\pi}{6}$ ($y = -3$)
Returns to midline at $x = \frac{5\pi}{3}$ ($y = -1$)
3

State the range

[-3, 1]

✓ Answer Range = $[-3, 1]$

Worked Example 3 — Counting Graphical Solutions

Stepwise

How many solutions does $2\cos x = 1$ have in $0 \leq x \leq 4\pi$?

1

Rewrite the equation

\cos x = \frac{1}{2}
2

Recall solutions in one period

In $[0, 2\pi]$, $\cos x = \frac{1}{2}$ has solutions at $x = \frac{\pi}{3}$ and $x = \frac{5\pi}{3}$. That's 2 solutions per period.
3

Count over the given domain

4\pi \text{ contains 2 periods} \Rightarrow 2 \times 2 = 4 \text{ solutions}

✓ Answer $4$ solutions

⚠️

Common Mistakes — Don't Lose Easy Marks

Mixing up reciprocal identities

Students sometimes write $\sec x = \frac{1}{\sin x}$ or $\csc x = \frac{1}{\cos x}$. Remember the pairs: sine-cosecant, cosine-secant, tangent-cotangent.

✓ Fix: "Secant goes with Cosine." Both start with "co-"? No — but memorise the pairs.

Forgetting quadrant signs when finding exact values

Even if you calculate the correct magnitude, you can lose marks if you give the wrong sign for the quadrant.

✓ Fix: ASTC — All positive in Q1, Sin in Q2, Tan in Q3, Cos in Q4.

Not factoring out $b$ when finding phase shifts

In $y = \sin(2x - \frac{\pi}{3})$, the phase shift is $\frac{\pi}{6}$, not $\frac{\pi}{3}$.

✓ Fix: Factor: $\sin\left(2\left(x - \frac{\pi}{6}\right)\right)$.

📓 Copy Into Your Books

▼

📖 Exact values

Know all special angles in both degrees and radians

🔢 Identities

$\sin^2 x + \cos^2 x = 1$
$1 + \tan^2 x = \sec^2 x$
$1 + \cot^2 x = \csc^2 x$

⚠️ Graph features

Sin/Cos: amplitude, period $2\pi$
Tan/Cot: asymptotes, period $\pi$
Phase shift = $c$ (right if positive)

💡 Modelling

$a = \frac{\text{max} - \text{min}}{2}$
$d = \frac{\text{max} + \text{min}}{2}$
$b = \frac{2\pi}{P}$

🧪 Activities

🔍 Activity 1 — Mixed Revision

Solve Each Problem

These questions draw on multiple topics from the module.

1 If $\cos \theta = -\frac{3}{5}$ and $\pi < \theta < \frac{3\pi}{2}$, find $\sin \theta$ and $\tan \theta$.

Type your answer:

Answer in your workbook.

Answer in your workbook
2 Simplify $\frac{1 - \cos^2 x}{\sin x \cos x}$.

Type your answer:

Answer in your workbook.

Answer in your workbook
3 State the amplitude, period, and range of $y = 4\cos(3x) + 2$.

Type your answer:

Answer in your workbook.

Answer in your workbook
4 How many solutions does $\sin x = -0.5$ have in $0 \leq x \leq 4\pi$?

Type your answer:

Answer in your workbook.

Answer in your workbook

Revisit Your Thinking

Earlier you were asked to list connections between the ideas in this module.

Here are some key connections:

The Pythagorean identity $\sin^2 x + \cos^2 x = 1$ comes directly from the unit circle equation $x^2 + y^2 = 1$.
A phase shift is just a horizontal translation of the basic graph.
Solving $\sin x = k$ is the same as finding where $y = \sin x$ and $y = k$ intersect.
Modelling uses the same transformations (amplitude, period, phase shift, vertical shift) that we use to sketch graphs.
Exact values, identities, and graphs all depend on the symmetries of the unit circle.

Now revisit your initial response. What connections did you identify? What new connections do you see now?

Look back at your initial response in your book. Annotate it with any new connections you've discovered.

Annotate your initial response in your book

Saved

Revisit Your Initial Thinking

Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?

Multiple Choice

5 random questions from a replayable lesson bank — feedback shown immediately

✍️ Short Answer

📝

Extended Questions

ApplyBand 4

8. Sketch $y = 3\sin\left(2x - \frac{\pi}{3}\right)$ for one complete period, starting from the first positive $x$-intercept. Label the amplitude, period, phase shift, and key points. 5 MARKS

Describe your sketch below:

Sketch in your workbook.

✏️ Sketch in your workbook

AnalyseBand 5

9. A water wheel rotates so that the height $h$ (in metres) of a bucket above the water surface is modelled by $h = 2\sin\left(\frac{\pi}{4}t\right) + 1$, where $t$ is in seconds. (a) Find the maximum and minimum heights. (b) Find the first time after $t = 0$ when the bucket is at a height of 2 metres. (c) State one limitation of using a simple sine model for this situation. 5 MARKS

Type your answer below:

Answer in your workbook.

✏️ Answer in your workbook

EvaluateBand 6

10. Evaluate the claim: "The graphs of $y = \sin x$ and $y = \cos x$ are identical except for a horizontal translation." Is this claim fully correct? Explain any limitations or exceptions. 3 MARKS

Type your answer below:

Answer in your workbook.

✏️ Answer in your workbook

✅ Comprehensive Answers

▼

🔍 Activity 1 — Mixed Revision Model Answers

1. In QIII, $\sin \theta = -\frac{4}{5}$, $\tan \theta = \frac{4}{3}$.

2. $\frac{\sin^2 x}{\sin x \cos x} = \frac{\sin x}{\cos x} = \tan x$.

3. Amplitude = 4, Period = $\frac{2\pi}{3}$, Range = $[-2, 6]$.

4. 4 solutions (2 per period, 2 periods).

❓ Multiple Choice

1. A — $\sin^2 x + \cos^2 x = 1$.

2. A — Amplitude = 2, period = $\frac{2\pi}{3}$.

3. A — $\tan x$ undefined at odd multiples of $\frac{\pi}{2}$.

4. A — Phase shift = $\frac{\pi}{6}$ right.

5. A — $a = \frac{10 - 2}{2} = 4$.

📝 Short Answer Model Answers

Q8 (5 marks): Rewrite: $y = 3\sin\left(2\left(x - \frac{\pi}{6}\right)\right)$ [1]. Amplitude = 3, Period = $\pi$, Phase shift = $\frac{\pi}{6}$ right [1]. First positive intercept at $x = \frac{\pi}{6}$ [1]. Max at $x = \frac{5\pi}{12}$ (value 3) [1]. Min at $x = \frac{11\pi}{12}$ (value $-3$) [1].

Q9 (5 marks): (a) Max = $2 + 1 = 3$ m, Min = $-2 + 1 = -1$ m [2]. (b) $2\sin\left(\frac{\pi}{4}t\right) + 1 = 2 \Rightarrow \sin\left(\frac{\pi}{4}t\right) = \frac{1}{2} \Rightarrow \frac{\pi}{4}t = \frac{\pi}{6} \Rightarrow t = \frac{2}{3}$ s [2]. (c) Limitation: the model doesn't account for friction, variable rotation speed, or the bucket dipping below the water surface [1].

Q10 (3 marks): The claim is correct for the standard functions: $\cos x = \sin\left(x + \frac{\pi}{2}\right)$ [1]. The graphs have the same shape, amplitude, and period, differing only by a horizontal shift of $\frac{\pi}{2}$ [1]. No exceptions for the basic functions, though transformed versions would need matching transformations [1].

Review and Connections

Download this lesson's worksheet

Key Facts

Concepts

Skills

Misconceptions to Fix

Module 2 at a Glance

Inquiry Question 1: Trigonometry

Inquiry Question 2: Trigonometric Functions and Graphs

Making Connections

From Identities to Graphs

From Equations to Graphs

From Graphs to Models

Worked Example 1 — Combined Identity and Exact Value

Worked Example 2 — Sketching a Transformed Graph

Worked Example 3 — Counting Graphical Solutions

Common Mistakes — Don't Lose Easy Marks

📓 Copy Into Your Books

📖 Exact values

🔢 Identities

⚠️ Graph features

💡 Modelling

Solve Each Problem

Multiple Choice

Extended Questions

✅ Comprehensive Answers

🔍 Activity 1 — Mixed Revision Model Answers

❓ Multiple Choice

📝 Short Answer Model Answers

Boss Battle — Trigonometry Final!

Mark lesson as complete