〰️ Trigonometric Functions
Angles turned into ratios — the mathematics behind every wave, rotation and oscillation an engineer will meet, from AC supplies to mechanism design.
2Topics
8Worked Examples
4Eng. Cases
12Practice Qs
📐Trig Ratiossin, cos, tan & the unit circle
🔗Identities & EquationsCompound angles, solving
⚙️ApplicationsReal engineering
🎮Quick-Fire GameTest your speed
📝PracticeGraded questions
🎯What you will learn
Define sin, cos and tan from the unit circle; apply the Pythagorean and compound-angle identities; and solve trigonometric equations over a given range, always checking for every solution, not just the first one a calculator returns.
💡 Radians aren't an alternative to degrees for style — they're the natural unit that makes calculus on trig functions work cleanly. Get comfortable switching between the two.
🧭How to use this resource
1. Read the rules banner at the top of each tab
2. Experiment with the interactive tool
3. Check the key vocabulary panel
4. Study the visual diagram, then the worked examples
📐
Trig Ratios & the Unit Circle
Every angle gives a unique sine, cosine and tangent — read straight off a circle of radius 1.
🔧Unit Circle ExplorerEnter an angle, then press Explore
Enter an angle in degrees, then press Explore to see sin, cos, tan and the identity check…
📖 Key Vocabulary
Radian — the natural angle unit; 2π rad = 360°.
Unit circle — a circle of radius 1 centred at the origin, used to define trig functions for any angle.
Period — the interval after which a trig function repeats (360° for sin, cos).
Quadrant — one of four regions of the circle; the sign of sin/cos/tan depends on which quadrant θ falls in.
👁️ Visual aid: cos θ is the horizontal shadow, sin θ is the vertical shadow of the radius line as it sweeps round the unit circle.
✏️Worked Example 1 — Reading ratios from a right triangleType: Basic ratio
Question A right triangle has opposite = 3, adjacent = 4, hypotenuse = 5. Find sinθ, cosθ, tanθ.
Apply definitions
sinθ = 3/5 = 0.6Continue
cosθ = 4/5 = 0.8, tanθ = 3/4 = 0.75Answer sinθ=0.6, cosθ=0.8, tanθ=0.75
🧭Worked Example 2 — Angles beyond 90° using the unit circleType: Quadrant awareness
Question Find sin(150°) and cos(150°) using the reference angle.
Reference angle
180° − 150° = 30°Quadrant 2 signs sin positive, cos negative in Q2
Answer sin(150°) = +sin(30°) = 0.5, cos(150°) = −cos(30°) = −0.866
📏Worked Example 3 — Converting degrees ⇄ radiansType: Unit conversion
Question Convert 135° to radians, and 2.5 rad to degrees.
Degrees→radians
135 × (π/180) = 3π/4 ≈ 2.356 radRadians→degrees
2.5 × (180/π) ≈ 143.2°Answer 135° ≈ 2.356 rad; 2.5 rad ≈ 143.2°
⚙️Worked Example 4 — Engineering context (mechanism angle)Type: Engineering
Question A crank arm 0.3 m long rotates from horizontal. Find the vertical height of its tip when the crank angle is 40°.
Model Height = r sinθ
Substitute
h = 0.3 × sin(40°) = 0.3 × 0.6428Answer h ≈ 0.193 m — the sine of the crank angle directly gives the vertical displacement.
🧠 Common slip: leaving your calculator in the wrong angle mode (degrees vs radians) — always check the mode before evaluating a trig function.
🔗
Identities & Solving Trigonometric Equations
Relationships that hold for every angle — the toolkit for simplifying expressions and solving equations.
🔧Trig Equation SolverSolve sinθ = k or cosθ = k over 0°–360°
Choose sin or cos, enter k (between −1 and 1), then press Solve to see both solutions in 0°–360°…
📖 Key Vocabulary
Identity — an equation true for all values of the variable, unlike an equation solved for specific values.
Reference angle — the acute angle between the terminal side and the x-axis; used to find all solutions.
Principal value — the single answer a calculator's sin⁻¹/cos⁻¹/tan⁻¹ returns.
General solution — a formula giving every possible solution, e.g. θ = 30°+360°n.
👁️ Visual aid: reading anticlockwise from Q1 gives "All Students Take Calculus" (A, S, T, C) — the mnemonic for which ratio is positive in each quadrant.
✏️Worked Example 1 — Using the Pythagorean identityType: Identity use
Question If sinθ = 0.6 and θ is acute, find cosθ.
Apply identity
cos²θ = 1 − sin²θ = 1 − 0.36 = 0.64Answer cosθ = √0.64 = 0.8 (positive, since θ is acute)
➕Worked Example 2 — Compound angle formulaType: Compound angle
Question Find sin(75°) using sin(45°+30°) and exact values.
Formula
sin(A+B) = sinAcosB + cosAsinBSubstitute
sin45cos30 + cos45sin30 = (√2/2)(√3/2) + (√2/2)(1/2)Answer = (√6+√2)/4 ≈ 0.9659
🔺Worked Example 3 — Solving a trig equation over a rangeType: Solve equation
Question Solve 2sinθ − 1 = 0 for 0° ≤ θ ≤ 360°.
Isolate
sinθ = 0.5Principal value
θ = sin⁻¹(0.5) = 30°Second solution sine is also positive in Q2:
θ = 180° − 30° = 150°Answer θ = 30° or 150°
⚙️Worked Example 4 — Engineering context (combining waves)Type: Engineering
Question Two AC voltages v₁ = 100sin(ωt) and v₂ = 100sin(ωt + 90°) combine. Find the resultant amplitude.
Expand v₂
100sin(ωt+90°) = 100cos(ωt)Combine
v = 100sin(ωt) + 100cos(ωt) = R sin(ωt+φ) where R = √(100²+100²)Answer R = 141.4 V — two equal signals 90° apart combine to √2 times one signal's amplitude.
🧠 Common slip: giving only the principal-value solution from a calculator and forgetting the second solution in the 0°–360° range that most equations have.
⚙️
Engineering Applications
Four live scenarios — each with its own calculator. Enter your own numbers.
🌊 AC Waveform — Instantaneous Value
An AC supply's instantaneous voltage is v = V_m sin(ωt + φ) — evaluate at any moment to find the live voltage.
v(t) = V_m sin(ωt + φ)
📟 Try it — Instantaneous Voltage
Enter amplitude, frequency, time and phase, then press Calculate…
📐 Resolving Forces — Components
A force at angle θ to the horizontal has components F cosθ (horizontal) and F sinθ (vertical) — essential for statics problems.
Fx = F cosθ, Fy = F sinθ
📟 Try it — Force Resolution
Enter force and angle, then press Calculate…
⚙️ Crank Mechanism — Slider Position
In a simple slider-crank, the slider's displacement from top-dead-centre is approximately x = r(1 − cosθ) for crank radius r.
x(θ) = r (1 − cosθ)
📟 Try it — Slider Displacement
Enter crank radius and angle, then press Calculate…
🔧 Vector Resultant — Two Forces at an Angle
The resultant of two forces meeting at angle θ uses the cosine rule: R² = F₁² + F₂² + 2F₁F₂cosθ.
R = √(F₁² + F₂² + 2F₁F₂cosθ)
📟 Try it — Resultant Force
Enter both forces and the angle between them, then press Calculate…
🎮
Quick-Fire Trigonometric Game
12 questions on ratios, identities and solving. How many can you get?
⚡ Trigonometric Challenge
Score: 0Question: 1/12Streak: 0🔥
Press Start to begin!
📝
Practice Questions
Graded from warm-up to challenge. Attempt fully before revealing the working.
Question 1 🥉2 marks
If cosθ = 0.8 (θ acute), find sinθ.
sin²θ = 1−0.64 = 0.36 → sinθ = 0.6
Question 2 🥉2 marks
Convert 60° to radians (in terms of π).
60 × π/180 = π/3
Question 3 🥉2 marks
Find tan(45°) exactly.
sin45/cos45 = 1 (equal legs) → tan45° = 1
Question 4 🥉2 marks
State the sign of sinθ and cosθ for θ in the third quadrant (180°–270°).
Q3 = "T" quadrant (CAST) — only tan positive
sinθ negative, cosθ negative
sinθ negative, cosθ negative
Question 5 🥈3 marks
Solve 2cosθ + 1 = 0 for 0°≤θ≤360°.
cosθ = −0.5
Reference angle 60°, cos negative in Q2 & Q3
θ = 120° or 240°
Reference angle 60°, cos negative in Q2 & Q3
θ = 120° or 240°
Question 6 🥈4 marks
Find cos(15°) using cos(45°−30°) and exact values.
cos(A−B)=cosAcosB+sinAsinB
=cos45cos30+sin45sin30
=(√2/2)(√3/2)+(√2/2)(1/2)
=(√6+√2)/4 ≈ 0.9659
=cos45cos30+sin45sin30
=(√2/2)(√3/2)+(√2/2)(1/2)
=(√6+√2)/4 ≈ 0.9659
Question 7 🥈3 marks
Resolve a 200 N force at 25° to the horizontal into its components.
Fx = 200cos25° = 181.3 N
Fy = 200sin25° = 84.5 N
Fy = 200sin25° = 84.5 N
Question 8 🥈4 marks
Solve tanθ = 1.5 for 0°≤θ≤360° (2 solutions).
Principal value: θ=tan⁻¹(1.5)=56.31°
tan also positive in Q3: θ=180+56.31
θ = 56.31° or 236.31°
tan also positive in Q3: θ=180+56.31
θ = 56.31° or 236.31°
Question 9 🥇4 marks
Prove sin(A+B) + sin(A−B) = 2sinAcosB using the compound angle formulae.
sin(A+B) = sinAcosB + cosAsinB
sin(A−B) = sinAcosB − cosAsinB
Add: cosAsinB terms cancel
Sum = 2sinAcosB ✓
sin(A−B) = sinAcosB − cosAsinB
Add: cosAsinB terms cancel
Sum = 2sinAcosB ✓
Question 10 🥇5 marks
A crank of radius 50 mm rotates. Find the slider displacement x=r(1−cosθ) at θ=120°.
x = 50(1−cos120°) = 50(1−(−0.5))
= 50(1.5)
x = 75 mm
= 50(1.5)
x = 75 mm
Question 11 🥇5 marks
Two forces 250 N and 180 N act at 70° to each other. Find the resultant using the cosine rule.
R² = 250²+180²+2(250)(180)cos70°
= 62500+32400+90000(0.342)
= 62500+32400+30777
R=√125677
R ≈ 354.5 N
= 62500+32400+90000(0.342)
= 62500+32400+30777
R=√125677
R ≈ 354.5 N
Question 12 🥇5 marks
Two AC signals v₁=150sin(ωt) and v₂=150sin(ωt+120°) combine. Find the resultant amplitude using the compound angle expansion.
v₂ = 150(sinωt cos120° + cosωt sin120°)
= 150(−0.5 sinωt + 0.866 cosωt)
v_total coefficient of sinωt: 150−75=75; of cosωt: 129.9
R=√(75²+129.9²)=√(5625+16874)
R ≈ 150 V (as expected — two equal-amplitude signals 120° apart sum to the same amplitude)
= 150(−0.5 sinωt + 0.866 cosωt)
v_total coefficient of sinωt: 150−75=75; of cosωt: 129.9
R=√(75²+129.9²)=√(5625+16874)
R ≈ 150 V (as expected — two equal-amplitude signals 120° apart sum to the same amplitude)