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Example: Without using a calculator, find cos(π − π ÷ 6).

Step 1 — Use the identity:
cos(π − θ) = −cosθ

Step 2 — Substitute θ = π ÷ 6:
cos(π − π ÷ 6) = −cos(π ÷ 6)

Step 3 — Use known value:
cos(π ÷ 6) = √3 ÷ 2

Final answer:
cos(π − π ÷ 6) = −√3 ÷ 2

🌍 Real-World Applications

• Simple harmonic motion (springs, pendulums) uses symmetric sine and cosine graphs.
• Electrical engineering (AC circuits) relies on cosine and sine wave symmetry to analyse voltage and current behaviour.
• Rotational motion in physics uses trig symmetries to determine positions over time.

📐 IA Spotlight

• Investigation of symmetry in periodic functions and its role in modelling real oscillations.
• Comparing sine vs cosine modelling for different physical systems (pendulum vs mass-spring).
• Exploring how symmetry simplifies solving trig equations in applied contexts.

🔍 TOK Perspective

• Trig graphs repeat infinitely, yet we summarise them with a small set of identities — what does this say about patterns in mathematics?
• How does the concept of symmetry influence the way mathematics describes natural phenomena?
• Are mathematical symmetries discovered in nature or invented as a framework for understanding it?

📝 Exam Tips

• Remember quadrant rules: sine stays positive in quadrants I and II, while cosine becomes negative in II.
• Use these symmetry identities to check answers quickly in multiple-choice questions.
• Correctly identifying the quadrant is the key step before applying any identity.

Example: Evaluate sin(75°) exactly.

Step 1 — Express 75° as a sum:
75° = 45° + 30°

Step 2 — Use the identity:
sin(45° + 30°) = sin45° × cos30° + cos45° × sin30°

Step 3 — Substitute exact values:
sin45° = √2 ÷ 2, cos45° = √2 ÷ 2
cos30° = √3 ÷ 2, sin30° = 1 ÷ 2

Step 4 — Compute:
sin75° = (√2 ÷ 2)(√3 ÷ 2) + (√2 ÷ 2)(1 ÷ 2)
sin75° = (√6 ÷ 4) + (√2 ÷ 4)
= (√6 + √2) ÷ 4

🌍 Real-World Applications

• AC electrical systems use compound angle identities to analyse phase shifts between voltage and current.
• GPS triangulation algorithms depend on compound angle relationships for accurate positioning.
• Wave interference patterns in physics rely heavily on angle-sum trigonometry.

📐 IA Spotlight

• Investigating how compound-angle identities appear in oscillations or resonance.
• Modelling two-source wave interference using sin(A ± B) expressions.
• Exploring GPS triangulation mathematics through angle sum relationships.

🔍 TOK Perspective

• Compound identities originate from geometric proofs. How does mathematical proof differ from scientific evidence?
• Why do different methods of proof (geometric, algebraic, analytic) all confirm the same identities?
• What does this reveal about the interconnected structure of mathematical knowledge?

📝 Exam Tips

• Always choose the simplest angle pair for evaluating expressions (e.g., 45° + 30°, not 60° + 15°).
• Keep exact values in radical form unless the question requests decimals.
• For tan(2A), check that tanA is defined—avoid values where tanA is undefined.

Example — solving using inverse functions

Solve sinθ = 0.6 for θ in the principal range.

θ = arcsin(0.6)

arcsin(0.6) returns an angle between −π ÷ 2 and π ÷ 2.
Using GDC: θ ≈ 0.6435 radians.

🧮 GDC Use

• Use inverse trig keys to compute arcsin, arccos and arctan values accurately.
• Graph reciprocal functions (secx, cosecx, cotx) to identify asymptotes and discontinuities.
• Use graph intersection methods to solve equations like secx = 2 or tanx = 3, by plotting the graph of the function and using x calc to find the answer.
• Always ensure the calculator is in radian mode for AHL trigonometry unless stated otherwise.

🌍 Real-World Applications

• Inverse trig functions appear frequently in navigation, engineering angles and direction calculations.
• secθ, cosecθ and cotθ are used in advanced mathematics and physics, including wave analysis and differentiation of trig functions.
• Computer graphics use arctan functions to determine camera angles and object orientation.

🔍 TOK Perspective

• Inverse functions illustrate how we decide which parts of a periodic function to treat as “principal” — is this a discovery or a convention?
• Reciprocal trig ratios highlight how mathematical structures extend logically — does mathematics grow from internal necessity or external usefulness?
• Different cultures contributed to trigonometric ideas; does this influence how we perceive the “facts” of mathematics?

📝 Paper Tips

• Always check whether the inverse trig answer must be in radians or degrees.
• Know the domain and range of arcsin, arccos and arctan — many mistakes come from giving the wrong branch.
• When simplifying trig expressions, use reciprocal identities to reduce complexity.
• Sketch a unit circle to understand signs (positive/negative) in different quadrants.

Example

For θ = 150°, find cosθ, sinθ and tanθ using the unit circle interpretation.

Step 1 — Convert to radians

150° = 150 × (π ÷ 180) = 5π ÷ 6

Step 2 — Determine coordinates

Angle 150° lies in quadrant II, where cosine is negative and sine is positive.
Coordinates on the unit circle: (−√3 ÷ 2, 1 ÷ 2)

Step 3 — Calculate tanθ

tanθ = sinθ ÷ cosθ = (1 ÷ 2) ÷ (−√3 ÷ 2) = −1 ÷ √3

Thus:
cosθ = −√3 ÷ 2
sinθ = 1 ÷ 2
tanθ = −1 ÷ √3

🧮 GDC Use

• Graph sinx, cosx and tanx to visualise intersections when solving equations.
• Use trace mode to identify exact or approximate values of θ where the graph equals a given number.
• Check ambiguous case solutions by graphing sinθ to see why two possible values of θ yield the same sine.
• Ensure the GDC is in radian mode for AHL topics unless specifically using degrees.

🌍 Real-World Applications

• Electrical engineering uses sinusoidal voltage and current, which follow the sine function exactly.
• Mechanical systems (rotors, gears, oscillators) use unit-circle-based rotational models.
• Computer graphics, wave simulation and signal processing use cosine-sine decomposition.

📐 IA Spotlight

• Investigate how the unit circle leads to periodic motion in mechanics or electricity.
• Model a real-life oscillation (pendulum, spring, alternating current) using sine or cosine functions.
• Explore the ambiguous case through geometric construction and compare exact vs approximate methods.

🔍 TOK Perspective

• The word “sine” has cultural roots in Indian and Arabic mathematics—does this show that mathematics is not culturally neutral?
• Unit-circle trigonometry evolved across multiple civilizations; how does mathematical knowledge develop through intercultural exchange?
• Many identities (e.g., cos²θ + sin²θ = 1) arise from geometry. Does this suggest mathematics is discovered rather than invented?

📝 Paper Tips

• Sketch the unit circle whenever possible—visualising quadrants helps avoid sign errors.
• Use radians unless the question specifically mentions degrees.
• For solving equations like sinx = a, always consider solutions in the full interval (e.g. 0 ≤ x ≤ 2π).
• For ambiguous case questions, consider the geometry—not just the algebraic sine rule.

Example

A circle has radius 7 cm. Find:

1. The arc length for an angle of 60°

2. The area of the corresponding sector.

Step 1 — Convert angle to radians

θ = 60° × (π ÷ 180) = π ÷ 3 radians

Step 2 — Arc length

s = rθ = 7 × (π ÷ 3)

s = 7π ÷ 3 cm

Step 3 — Area of sector

A = (1 ÷ 2) × r² × θ

A = (1 ÷ 2) × 49 × (π ÷ 3)

A = 49π ÷ 6 cm²

🧮 GDC Use

• Use “angle conversion” functions to switch between radians and degrees quickly.
• Graphing radian-based trigonometric functions is more accurate; ensure the calculator mode is set to radians when needed.
• For arc length and sector area, enter formulas directly to check working and avoid arithmetic errors.

🌍 Real-World Applications

• Circular motion in physics uses radians to measure angular displacement, velocity and acceleration.
• Wave motion and diffraction patterns rely on radian measure due to their trigonometric foundations.
• Engineering fields such as robotics and mechanical systems measure rotation using radians for precision.

📐 IA Spotlight

• Explore arc lengths or sector areas in real objects (gears, wheels, tracks) and analyse how angle measurement affects calculations.
• Investigate circular motion and model angular velocity or acceleration using radian-based trigonometric functions.
• Analyse why radians simplify calculus and compare results obtained using degrees vs radians.

🔍 TOK Perspective

• Radians arise from geometry, while degrees are human conventions. Which system provides more “mathematical truth”?
• Is a natural measure (rad) “better” than a human-made one (deg)? By what criteria do mathematicians decide?
• Different cultures used different angle systems—Babylonians used 360 degrees. Does the choice of unit affect how mathematics is understood?

📝 Paper Tips

• Check calculator mode — most mistakes occur from using degree mode in radian questions.
• Sector formulas only work when θ is in radians.
• Leave answers as exact multiples of π unless the question asks for decimals.
• Show conversion steps clearly to earn method marks.

Example — equation of a Voronoi edge

Two sites are P(2, 1) and Q(8, 5). Find the equation of the boundary between their Voronoi cells.

Midpoint:

M = ((2 + 8) ÷ 2, (1 + 5) ÷ 2) = (10 ÷ 2, 6 ÷ 2) = (5, 3)

Gradient of P Q:

m = (5 − 1) ÷ (8 − 2) = 4 ÷ 6 = 2 ÷ 3

Gradient of perpendicular bisector:

m_⊥ = −1 ÷ (2 ÷ 3) = −3 ÷ 2

Equation through M(5, 3):

y − 3 = (−3 ÷ 2) × (x − 5)
This line forms the Voronoi edge between P and Q.

🧮 GDC Use

• Plot the sites as points to visualise the Voronoi cells. Some GDCs or graphing apps support drawing perpendicular bisectors directly.
• Use distance functions to check which site is closest to a given point by computing several distances and comparing them.
• For the “toxic waste dump” problem, store candidate vertex coordinates and evaluate distances to each town using the distance formula on the GDC.

🌍 Real-World Connections

• Urban planning: Assigning each house to its nearest school, hospital or fire station.
• Spread of diseases: Modelling which health centre is likely to serve each area, or which outbreak source is nearest.
• Ecology and meteorology: Dividing a region according to the nearest weather station or sampling site to estimate rainfall, temperature or species density.
• Telecommunications and computing: Cell tower coverage areas, clustering in data science and nearest neighbour search algorithms.

📐 IA Spotlight

• Use real locations from your city (schools, clinics, bus stops) and create a Voronoi diagram on graph paper or using technology. Analyse which households are closest to which facility.
• Investigate how adding a new site (for example, a new clinic) changes the sizes of Voronoi cells and shifts boundaries of service areas.
• Explore relationships between cell area and underlying data like population or rainfall, discussing limitations of nearest neighbour interpolation.

🔍 TOK Perspective

• Voronoi diagrams show how we divide continuous space into discrete regions based on a chosen rule (nearest site). To what extent is this division a natural feature of the world, and to what extent is it a human-made model?
• Different disciplines (geography, biology, economics, computer science) use the same mathematical structure. Does this support the idea that mathematics provides a shared language across areas of knowledge?

📝 Paper Tips

• Label sites clearly and sketch a rough Voronoi diagram if not provided; this helps identify which region a point belongs to.
• When asked for a boundary equation, remember it is a perpendicular bisector, so find midpoint and negative reciprocal gradient carefully.
• For “nearest site” questions, compare squared distances when possible to avoid unnecessary square roots; the smallest squared distance gives the nearest site.
• In “toxic waste dump” questions, focus on vertices where three regions meet and compare their distances to all sites systematically.

Example

Find the perpendicular bisector of the segment joining A(2, 3) and B(8, 9).

Step 1: Midpoint

Midpoint = ((2 + 8) ÷ 2, (3 + 9) ÷ 2) = (10 ÷ 2, 12 ÷ 2) = (5, 6)

Step 2: Gradient of AB

m = (9 − 3) ÷ (8 − 2) = 6 ÷ 6 = 1

Step 3: Gradient of perpendicular bisector

m_⊥ = −1 ÷ 1 = −1

Step 4: Equation

Use point-slope form with midpoint (5, 6):
y − 6 = −1 × (x − 5)

Expand:
y − 6 = −x + 5
y = −x + 11

Perpendicular bisector: y = −x + 11

Example

A line segment has midpoint M(4, −2). The gradient of the segment is 2.
Find the equation of the perpendicular bisector.

Step 1: Perpendicular gradient

m_⊥ = −1 ÷ 2 = −0.5

Step 2: Equation through midpoint

y + 2 = −0.5 × (x − 4)

y + 2 = −0.5x + 2
y = −0.5x

Perpendicular bisector: y = −0.5x

🧮 GDC Use

• You can use the GDC to check midpoints and gradients quickly by plugging values into tables or geometry tools.
• Graph both the segment AB and your perpendicular bisector to visually confirm the line crosses the midpoint and is at 90°.
• Useful especially in exam settings to verify your negative reciprocal gradient and avoid sign errors.

🌍 Real-World Applications

• Used in locating equidistant points between two transmitters, towers or landmarks.
• Base method in triangulation and surveying for mapping and positioning.
• Essential for finding the centre of a circle formed by three non-collinear points.
• Appears in design and engineering when constructing perpendicular supports or symmetrical shapes.

🔍 TOK Perspective

• The perpendicular bisector represents a set of points defined by a logical rule: “equidistant from two points”. Is this mathematical locus an invention or a discovery?
• Coordinate geometry allows us to express geometric ideas algebraically. Does representing a geometric idea in algebra change how we understand it?

📝 Paper Tips

• Write the midpoint first—it anchors the perpendicular bisector.
• Always double-check the perpendicular gradient: it must be the negative reciprocal.
• Simplify the equation neatly and give final answer in the form y = mx + c unless instructed otherwise.
• A sketch (even small) helps prevent sign mistakes with gradients.

Example

A circle has radius 7 cm. A sector has central angle 60°. Find:

1. Arc length
   Arc = (θ ÷ 360) × 2πr
   = (60 ÷ 360) × 2π × 7
   = (1 ÷ 6) × 14π
   = (14π ÷ 6) = (7π ÷ 3) cm
   ≈ 7.33 cm.
2. Sector area
   Area = (θ ÷ 360) × πr²
   = (60 ÷ 360) × π × 7²
   = (1 ÷ 6) × 49π
   = 49π ÷ 6 cm²
   ≈ 25.66 cm².

🧮 GDC Use

• Store r, θ and π values in your calculator memory when using them repeatedly in one question.
• Use the π key rather than 3.14 to keep results accurate and consistent.
• For multi-step questions (for example when the arc length is then used to work out something else), keep full precision on the GDC and round only in your final written answer.

🌍 Real-World Connections

• Designing curved paths or tracks, where the arc length gives the distance travelled along a bend.
• Calculating the area of circular sectors in engineering, for example the cross-section of partially opened valves or gates.
• Determining the size of slices in circular charts or pie charts, where the central angle represents a proportion of the whole.

📐 IA Spotlight

• Investigate how changing the angle of a sector affects arc length and area while keeping radius fixed, and discuss which changes are linear and which are not.
• Model real circular objects (for example, pizzas, wheels, circular gardens) and compare theoretical sector areas with measurements or photographs.

🔍 TOK Perspective

• The idea that a sector with angle θ represents θ ÷ 360 of a full circle relies on the convention that a full turn is 360°. Why 360 and not another number?
• Our experience of circular motion (wheels, rotations, cycles) influences how we define and use angles. To what extent is this mathematical or cultural?

Illustrative scenario — combining Pythagoras and trig

A surveyor stands 40 metres from the base of a building on flat ground.
The angle of elevation to the top is 32°. The roof has a short vertical antenna whose top is 5 metres higher than the roof itself.

Step one is to model the building without the antenna using a right-angled triangle: ground, building height, and line of sight.
Using tan 32° = height ÷ 40, the height of the building alone is 40 × tan 32°.
The total height including the antenna is then that value plus 5 metres.

The exact calculator evaluation is less important here than the reasoning:
the triangle lets us express an unreachable vertical distance in terms of a measurable horizontal distance and a measurable angle.

🧮 GDC Use

• Use the calculator’s trig functions to evaluate expressions like 40 × tan 32° or to compute inverse trigonometric functions quickly.
• When solving multi-step problems, store intermediate values in memory (for example, the height found from one triangle that is used in a second triangle) to avoid rounding errors.
• Some GDCs can draw simple triangles or show values in a table, which can be used to check whether your answers for heights or distances are reasonable.
• Always show the algebraic setup in your written work; the GDC is for numerical evaluation, not for replacing reasoning.

🌍 Real-World Connections

• Triangulation and surveying: Used to map coastlines, mountain ranges and city layouts by measuring angles from known baseline distances.
• Navigation and aviation: Pilots and ship captains use bearings and angles of elevation or depression when approaching runways or navigating around hazards.
• Radio and satellite communication: Positioning of radio towers and satellite dishes relies on angles and heights to maximise signal strength.
• Parallax methods in astronomy: Distances to nearby stars can be estimated using tiny angles and Earth’s orbital radius as the baseline.

📐 IA Spotlight

• Investigate the accuracy of height measurements found using angles of elevation from different distances away from a building or tree.
• Design a small triangulation project in your local area, using multiple observation points and bearings to estimate the position of an inaccessible point.
• Compare measurements obtained using trigonometry with those from digital tools such as phone range-finder apps or mapping software, and analyse discrepancies.

🌐 EE Focus

• Explore the historical development of triangulation and its role in determining the size and curvature of the Earth, including early geodetic surveys.
• Examine different proofs and generalisations of Pythagoras’ theorem, including those that apply in non-Euclidean geometries where the angle sum of a triangle is not 180°.
• Investigate how modern GPS systems conceptually rely on distance and angle ideas, and compare this with classical triangulation methods.

🔍 TOK Perspective

• In Euclidean geometry the angles of a triangle add to 180°, but on curved surfaces this is no longer true. What does this say about the status of mathematical truths?
• When explorers used trigonometry to estimate Earth’s size, how did they decide whether the models and measurements were “good enough” to count as knowledge?
• Many cultures contributed to trigonometry and proofs of Pythagoras’ theorem. How does recognising this global history affect our view of ownership and authorship in mathematics?

❤️ CAS Ideas

• Organise a “geometry walk” around your school, where participants estimate heights or widths of buildings and trees using simple trigonometric tools.
• Create an interactive workshop for younger students on bearings and map-reading, using small navigation challenges in the playground or local park.
• Collaborate with geography or physics teachers to design a field study that measures slopes, distances and directions in a nearby natural area.

📝 Paper 1 and Paper 2 Tips

• Start every problem with a neat, labelled diagram. Most mistakes in this topic come from missing or mis-labelled angles and distances.
• Clearly state which method you are using: “Using Pythagoras”, “Using sin rule”, “Using cosine rule”, or “Using right-angled trig and tan”.
• Keep angle mode in degrees unless the question explicitly uses radians, and check that answers are reasonable (for example, a height should not be less than a horizontal distance if the angle of elevation is large).
• Round only at the end of your solution, especially if one result is reused in a later calculation.

🧠 Examiner Tip

Examiners report that many students try to jump straight to formulas without first understanding the situation.
Marks are often lost because of an incorrect diagram or misidentified angle.
If you take time to sketch, label and think about the story the question is telling, the trigonometry becomes routine and you gain both method and accuracy marks.

📐 IA Spotlight

• Investigate triangulation methods to estimate heights or distances you cannot measure directly (for example, height of a building, width of a river) using sine and cosine rules.
• Collect real data using a clinometer or phone sensor and compare theoretical distances against measured ones, discussing sources of error.

🌐 EE Focus

• Explore the history and development of trigonometry in ancient cultures (Indian, Chinese, Greek, Islamic) and how triangle methods evolved.
• Study error propagation when using trig rules to approximate long distances or heights in navigation and surveying.

Worked example 1 — finding a side

A ladder leans against a wall, making an angle of 70° with the horizontal.
The ladder is 5 metres long. Find the height the ladder reaches on the wall.

1. The ladder is the hypotenuse. The height on the wall is opposite the 70° angle.
2. Use sin: sin 70° = opposite/hypotenuse.
3. Let h be the height. Then sin 70° = h/5.
4. So h = 5 × sin 70°.
5. Using calculator: sin 70° ≈ 0.9397.
   h ≈ 5 × 0.9397 ≈ 4.70 m.

Worked example 2 — finding an angle

In a right-angled triangle, the side opposite θ is 7 cm and the adjacent side is 10 cm. Find θ to the nearest degree.

1. Known: opposite and adjacent → use tan.
2. tan θ = opposite / adjacent = 7/10 = 0.7.
3. θ = tan⁻¹(0.7) ≈ 34.99°.
4. To nearest degree, θ ≈ 35°.

🧮 GDC Tips

• Check your calculator angle mode (degrees for IB SL questions unless stated otherwise).
• Use the built-in sin⁻¹, cos⁻¹, tan⁻¹ functions to find angles from ratios.
• Store intermediate results to avoid rounding too early when using values again (for example, use the answer memory).

Worked example 3 — sine rule to find a side

In triangle ABC, A = 40°, B = 75°, and side a (opposite A) is 8 cm. Find side b.

1. Use a / sin A = b/sin B.
2. 8/sin 40° = b/sin 75°.
3. Rearrange: b = sin 75° × (8/sin 40°).
4. Compute: sin 75° ≈ 0.9659; sin 40° ≈ 0.6428.
5. b ≈ 0.9659 × (8/0.6428) ≈ 0.9659 × 12.44 ≈ 12.02 cm.

📝 Paper 1 & Paper 2 Tips

• Always draw a clear, labelled diagram with angles and sides marked before choosing a rule.
• Write down the formula first, then substitute values — this gains method marks even if arithmetic slips occur.
• Round angles at the very end; use unrounded values in later calculations.
• Check whether your answer is reasonable: angles should add to about 180°; side opposite larger angle should be longest.

Worked example 5 — using cosine rule to find a side

In triangle X Y Z, sides adjacent to angle Z are 7 cm and 10 cm, and angle Z is 50°. Find side z opposite angle Z.

1. Let a = 7, b = 10, C = 50°, c = z.
2. c² = a² + b² − 2 × a × b × cos C.
3. c² = 7² + 10² − 2 × 7 × 10 × cos 50°.
4. c² = 49 + 100 − 140 × cos 50°.
5. cos 50° ≈ 0.6428 → 140 × 0.6428 ≈ 89.99.
6. c² ≈ 149 − 89.99 ≈ 59.01.
7. c ≈ √59.01 ≈ 7.68 cm.

Worked example 6 — using cosine rule to find an angle

Triangle A B C has sides a = 6 cm, b = 8 cm, c = 9 cm. Find angle C to the nearest degree.

1. Use cos C = (a² + b² − c²) / (2 × a × b).
2. cos C = (6² + 8² − 9²) / (2 × 6 × 8).
3. cos C = (36 + 64 − 81) / 96 = 19 / 96 ≈ 0.1979.
4. C = cos⁻¹(0.1979) ≈ 78.6° → C ≈ 79°.

Worked example 7 — area from two sides and an included angle

A triangle has sides 9 cm and 12 cm with an included angle of 40°. Find its area.

1. Area = (1/2) × 9 × 12 × sin 40°.
2. (1/2) × 9 × 12 = 54.
3. sin 40° ≈ 0.6428.
4. Area ≈ 54 × 0.6428 ≈ 34.71 cm².

🔍 TOK Perspective

Trigonometry is based on axioms and definitions about triangles and angles. How universal are these ideas?
When explorers used trigonometry to estimate the size and shape of Earth, how did assumptions and measurement tools influence what was accepted as knowledge?

❤️ CAS Ideas

• Run a “map-making” or orienteering activity for younger students where they locate points by measuring angles and distances.
• Create a mural or digital artwork based on triangular patterns, labelling the trig relationships used.

🧠 Examiner Tip

Examiners frequently see answers where students select the wrong rule or forget to indicate which side corresponds to which angle.
Always state clearly: “using the sine rule” or “using the cosine rule” and match labels (a opposite A, b opposite B, c opposite C). Clear diagrams and formula statements are an easy way to secure method marks.