Pure 1 — Trigonometry | Full Lesson

Pure Mathematics 1

Trigonometry Lesson

Welcome. Trigonometry is the study of triangles and angles. It might look intimidating, but I promise you — it all comes down to three simple ratios. By the end you will understand the unit circle, exact values, graphs, identities, and how to solve trig equations. Let us begin.

What is Trigonometry?

Trigonometry is all about the relationship between angles and side lengths in triangles. We start with right-angled triangles — triangles that have one 90° angle.

Every right-angled triangle has three sides:

Hypotenuse (hyp) — the longest side, opposite the right angle

Opposite (opp) — the side opposite the angle we are looking at

Adjacent (adj) — the side next to the angle (but not the hypotenuse)

SOH CAH TOA sin θ = opposite / hypotenuse | cos θ = adjacent / hypotenuse | tan θ = opposite / adjacent

This is the single most important thing to remember in all of trigonometry. SOH CAH TOA gives you the three basic ratios.

Teacher's Tip: Learn SOH CAH TOA as a word. Say it out loud: "Soak-a-toe-ah." Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj. Write it on your hand for the first week.

Example 1: A right-angled triangle has opposite = 3, adjacent = 4, hypotenuse = 5. Find sin θ, cos θ, tan θ.

sin θ = opp/hyp = 3/5 = 0.6

cos θ = adj/hyp = 4/5 = 0.8

tan θ = opp/adj = 3/4 = 0.75

Answer: sin θ = 0.6, cos θ = 0.8, tan θ = 0.75

Example 2: In a triangle, the hypotenuse is 10 cm and the angle is 30°. Find the length of the opposite side.

We know hyp = 10, θ = 30°, and we want opp. Which ratio uses opp and hyp?

SOH: sin θ = opp / hyp

sin 30° = opp / 10

opp = 10 × sin 30°

sin 30° = 0.5

opp = 10 × 0.5 = 5

Answer: The opposite side is 5 cm

Example 3: A ladder 6 m long leans against a wall at 60° to the ground. How high up the wall does it reach?

The ladder is the hypotenuse (6 m). The height up the wall is opposite the 60° angle.

sin 60° = opp / 6

opp = 6 × sin 60° = 6 × (√3 / 2) = 3√3

Answer: 3√3 m (about 5.2 m)

The Unit Circle

The unit circle is a circle of radius 1 centred at the origin. It is the most powerful tool in trigonometry because it lets us define sin, cos, and tan for any angle, not just acute ones.

For an angle θ measured anticlockwise from the positive x-axis:

🔵 sin θ is the y-coordinate of the point on the circle

🟢 cos θ is the x-coordinate of the point on the circle

🟡 tan θ = y/x = sin θ / cos θ

Unit Circle Definitions cos θ = x | sin θ = y | tan θ = y / x (x ≠ 0)

Drag the slider below to see how the angle changes the coordinates.

θ = 0°

sin θ = 0.000

cos θ = 1.000

tan θ = 0.000

Angles in All Four Quadrants

The unit circle is divided into four quadrants:

Quadrant I (0° to 90°): sin +, cos +, tan +

Quadrant II (90° to 180°): sin +, cos −, tan −

Quadrant III (180° to 270°): sin −, cos −, tan +

Quadrant IV (270° to 360°): sin −, cos +, tan −

A useful mnemonic: "All Students Take Coffee" — going anticlockwise from QI, the functions that are positive are: All, Sin, Tan, Cos.

Teacher's Tip: The unit circle is worth more than memorising formulas. If you understand the unit circle, you understand trigonometry. Spend time here.

Exact Values

Some angles appear so often that you must memorise their sin, cos, and tan values exactly (as fractions and surds). These are called exact values.

Click each card below to reveal the value — test yourself!

The Pattern to Memorise

For sin of 0°, 30°, 45°, 60°, 90° — think of the sequence:

sin 0° = √0 / 2 = 0

sin 30° = √1 / 2 = 1/2

sin 45° = √2 / 2

sin 60° = √3 / 2

sin 90° = √4 / 2 = 1

For cos, reverse the sequence: cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0.

For tan, use tan θ = sin θ / cos θ.

Teacher's Tip: The "√0, √1, √2, √3, √4 over 2" trick for sin is the fastest way to memorise exact values. Write it out five times and you will never forget it.

Graphs of sin, cos, tan

The graphs of trigonometric functions repeat forever — they are periodic. Understanding their shapes is essential.

The Sine Graph: y = sin x

📏 Shape: Smooth wave starting at 0. Goes up to 1, down to −1.

🔁 Period: 360° (or 2π radians) — repeats every 360°

📐 Amplitude: 1 (distance from centre to peak)

🎯 Key points: (0,0), (90°,1), (180°,0), (270°,−1), (360°,0)

The Cosine Graph: y = cos x

📏 Shape: Same wave shape, but shifted — starts at 1.

🔁 Period: 360° (2π)

📐 Amplitude: 1

🎯 Key points: (0°,1), (90°,0), (180°,−1), (270°,0), (360°,1)

The Tangent Graph: y = tan x

📏 Shape: Repeats every 180°, has asymptotes where cos x = 0.

🔁 Period: 180° (π radians)

🎯 Key points: (0°,0), (45°,1), (90°, undefined), (135°,−1), (180°,0)

Trig Graph Features Period of sin and cos = 360° (2π) | Period of tan = 180° (π) | Amplitude = 1

Trigonometric Identities

Identities are equations that are true for every value of θ. They let you rewrite trig expressions in different forms.

Identity 1 — The Pythagorean Identity sin²θ + cos²θ = 1

This comes from the unit circle: x² + y² = 1, and since x = cos θ, y = sin θ, we get sin²θ + cos²θ = 1.

Identity 2 — Tangent Identity tan θ = sin θ / cos θ

This follows directly from tan θ = y/x and the unit circle definitions.

Example 4: If sin θ = 3/5 and θ is acute, find cos θ and tan θ.

We know: sin²θ + cos²θ = 1

(3/5)² + cos²θ = 1

9/25 + cos²θ = 1

cos²θ = 1 − 9/25 = 16/25

cos θ = 4/5 (positive since θ is acute)

tan θ = sin θ / cos θ = (3/5) / (4/5) = 3/4

Answer: cos θ = 4/5, tan θ = 3/4

Example 5: Show that 1 − cos²θ = sin²θ.

Start from the identity: sin²θ + cos²θ = 1

Subtract cos²θ from both sides:

sin²θ = 1 − cos²θ

Therefore 1 − cos²θ = sin²θ ✓

True for all θ

Solving Trigonometric Equations

Solving a trig equation means finding all angles that satisfy the equation within a given range. Because trig functions repeat, there are usually two solutions in each period.

The CAST Diagram Method

The CAST diagram (also called the quadrant diagram) helps you find all solutions:

CAST — Which functions are positive Quadrant II (S): sin + | Quadrant I (A): all +
Quadrant III (T): tan + | Quadrant IV (C): cos +

To solve sin θ = k:

1. Find the principal value (use calculator or exact value) — call it α

2. sin is positive in QI and QII, so solutions are θ = α and θ = 180° − α

3. Add multiples of 360° for further solutions

Example 6: Solve sin θ = 0.5 for 0° ≤ θ < 360°

Step 1: Principal value: sin⁻¹(0.5) = 30°

Step 2: sin is positive in QI and QII.

QI solution: θ = 30°

QII solution: θ = 180° − 30° = 150°

Step 3: No further solutions in range (adding 360° would go outside).

Answer: θ = 30° or 150°

Example 7: Solve cos θ = −1/√2 for 0° ≤ θ < 360°

Step 1: cos is negative, so ignore the sign for the principal value.

cos⁻¹(1/√2) = 45° (this is our reference angle α)

Step 2: cos is negative in QII and QIII.

QII: θ = 180° − 45° = 135°

QIII: θ = 180° + 45° = 225°

Answer: θ = 135° or 225°

Example 8: Solve tan θ = √3 for 0° ≤ θ < 360°

Step 1: tan⁻¹(√3) = 60°

Step 2: tan is positive in QI and QIII.

QI: θ = 60°

QIII: θ = 180° + 60° = 240°

(Add 180° for the next tan solution)

Answer: θ = 60° or 240°

Using the Graph to Solve

Alternatively, draw the graph and find where it crosses the horizontal line y = k. The intersection points are your solutions.

Transformations of Trig Graphs

The general transformed sine wave is:

Transformed Sine Graph y = a sin(bx + c) + d

Each parameter changes something specific:

🔵 a — Amplitude = |a|. The graph stretches vertically by a factor of a. If a is negative, the graph is reflected in the x-axis.

🟢 b — Period = 360° / |b| (or 2π / |b| in radians). A larger b means more cycles in the same space.

🟡 c — Phase shift (horizontal shift). The graph shifts left by c/b (if c > 0, shift left; if c < 0, shift right).

🟠 d — Vertical shift. The whole graph moves up by d (or down if d is negative).

Use the sliders below to see how each parameter transforms the sine wave.

a = 1 b = 1 c = 0 d = 0

y = sin x

Practice — Step by Step

Solve each question one step at a time. Type your answer for each step and click Check Step. You must get each step correct before moving to the next.

Mastery Kit

Mathematics is the art of giving the same name to different things. Trigonometry is where that art really shines — angle, ratio, wave, circle — all the same dance.

— Henri Poincaré (adapted)

⚡ Quick Hacks

CAST diagram trick: Draw a cross (quadrants). Label them anticlockwise: A (all), S (sin), T (tan), C (cos). "All Students Take Coffee." This tells you which functions are positive in each quadrant.
Exact values pattern: For sin: √0/2, √1/2, √2/2, √3/2, √4/2 from 0° to 90°. Reverse for cos. This is the easiest hack in all of A-Level maths.
Sine and cosine graphs: Remember "sin starts at 0, cos starts at 1." If you forget which is which, check the value at 0°.
Tan is sin/cos: Whenever stuck on tan, rewrite it as sin/cos and work from there.
Solving equations: Find the principal value first, then use CAST to find the other solution(s) in the range.

📌 Key Notes

SOH CAH TOA: The foundation of all trigonometry. sin = opp/hyp, cos = adj/hyp, tan = opp/adj.
Unit circle: x = cos θ, y = sin θ. The radius is 1, so x² + y² = 1.
Exact values: Memorise sin/cos/tan for 0°, 30°, 45°, 60°, 90°. Use the √0/2 through √4/2 trick.
Periodicity: sin and cos repeat every 360° (2π). tan repeats every 180° (π).
Identities: sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ are your two essential tools.
CAST diagram: Use it to find all solutions of a trig equation in a given range.

⚠️ Common Mistakes

Calculator in wrong mode: Always check whether you need degrees or radians. This is the single most common trig mistake.
Only one solution: Trig equations usually have two solutions in 0°–360°. Using CAST prevents this mistake.
Forgetting tan asymptotes: tan 90° and tan 270° are undefined (division by zero). The graph has vertical asymptotes there.
Wrong quadrant sign: cos 120° is negative (−1/2), not positive. Use the unit circle to check signs.
Mixing up sin and cos graphs: sin starts at 0, cos starts at 1. Draw the key points from 0° to 360° to get the shape right.
Forgetting the ± in cos⁻¹: When solving cos θ = k, the second solution is 360° − α (not 180° − α like sin).

🧠 Memory Aid — Trig Cheat Sheet

SOH CAH TOA: sin=opp/hyp, cos=adj/hyp, tan=opp/adj

Unit circle: cos θ = x, sin θ = y

Identity: sin²θ + cos²θ = 1 | tan θ = sin θ / cos θ

Exact sin: √n/2 for n = 0,1,2,3,4 at 0°,30°,45°,60°,90°

y = a sin(bx + c) + d: |a| = amp, 360°/|b| = period, shift = −c/b, vert = d

💡

The Golden Rule of Trigonometry

When in doubt, draw the unit circle. It shows you the signs, the values, and the relationships. The circle is your compass — and your anchor — in every trig problem.

You Have Completed the Trigonometry Lesson

Here is what you have learned:

✅ SOH CAH TOA — sin, cos, tan as ratios in right-angled triangles
✅ The unit circle — sin θ = y, cos θ = x, tan θ = y/x, all four quadrants
✅ Exact values for 0°, 30°, 45°, 60°, 90° (and their radian equivalents)
✅ Graphs of sin, cos, tan — shape, period, amplitude, key points
✅ Trigonometric identities — sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ
✅ Solving trig equations using the CAST diagram and graphs
✅ Transformations — y = a sin(bx + c) + d: amplitude, period, phase shift, vertical shift

Well done. You now have a solid foundation in trigonometry. This is one of the most beautiful and useful areas of mathematics — and you have mastered it.