Unit 5: Trigonometry, Seating and Coded Inequalities

I. Orientation: Analytical Framework

This unit combines mathematical modelling with logical arrangement and comparison. Trigonometry converts angles and distances into measurable relationships; seating problems convert verbal clues into spatial positions; coded inequalities translate symbolic statements into ordered relationships between quantities.

• Core principle: Represent the given information in a precise mathematical or visual form before drawing conclusions.
• Trigonometric convention: Unless stated otherwise, objects are vertical, ground is horizontal, and lines of sight are straight.
• Angular convention: Angles of elevation and depression are measured from a horizontal line.
• Seating convention: In linear arrangements, left and right depend on the direction faced; in circular arrangements, they depend on whether people face the centre or outside.
• Inequality convention: An inequality states an order relation, such as (a>b), while a coded inequality replaces the familiar sign with an assigned symbol.
• Equation comparison convention: Roots should be found or bounded accurately before comparing the variables represented by them.
• Preferred workflow:
  • Identify known and unknown quantities.
  • Draw a diagram or decode the notation.
  • Apply only the relevant rule.
  • Check every condition before selecting the final relationship.

II. Trigonometric Applications — Lines of Sight and Motion

Trigonometric applications use right-angled triangles to connect an observable angle with an unknown height, horizontal distance, or changing position. The primary ratios are:

sin θ = P/H
cos θ = B/H
tan θ = P/B

Here, (\theta) is the acute angle, (P) is the side perpendicular to (\theta), (B) is the adjacent or base side, and (H) is the hypotenuse.

A. Problems based on height and distance

Height-and-distance problems model a vertical object, horizontal ground, and line of sight as a right-angled triangle.

• Angle of elevation: When an observer looks upward, the angle between the horizontal and the line of sight is the angle of elevation.
• Angle of depression: When an observer looks downward, the angle between the horizontal and the downward line of sight is the angle of depression.
• Equality of angles: Horizontal lines are parallel, so the angle of depression from the top equals the corresponding angle of elevation from the ground.
• Basic height relation:

tan θ = h/d
h = d tan θ
d = h/tan θ

Here, (h) is vertical height, (d) is horizontal distance, and (\theta) is the angle of elevation or depression.

• Observer’s height: If the observer’s eye is (e) units above the ground, trigonometry gives the height above eye level. Thus:

Total height = e + d tan θ

Here, (e) is the observer’s eye height and (d) is the horizontal distance from the object.

• Two observation points: If observations are made from two points on the same straight line, assign separate horizontal distances and connect them using the known separation.
• Shadow problems: A vertical object and its shadow form a right triangle; the Sun’s elevation (\theta) gives (h=s\tan\theta), where (s) is shadow length.
• Units: Height and horizontal distance must use compatible units, such as metres and metres.
• Worked example: A person stands (20) m from a tower and observes its top at (45^\circ). Ignoring eye height:

h = 20 tan 45°
  = 20 × 1
  = 20 m

Therefore, the tower is (20) m high.

• Common limitation: The model becomes unreliable if the ground is sloping or the object is not vertical unless those deviations are explicitly included.

B. Moving object based problems

Moving-object problems combine trigonometric position relationships with the distance–speed–time formula.

• Motion relation:

Distance = Speed × Time
s = vt

Here, (s) is distance travelled, (v) is constant speed, and (t) is elapsed time.

• Changing angle: As an object approaches a tower, the horizontal distance decreases and the angle of elevation increases; as it moves away, the reverse occurs.
• Fixed height model: For an object observed from the top of a tower of height (h):

d = h/tan θ

Here, (d) is the object’s horizontal distance from the tower and (\theta) is the angle of depression.

• Distance travelled: If the initial and final distances are (d_1) and (d_2), an approaching object travels (d_1-d_2); an object crossing the reference point may require (d_1+d_2).
• Worked example: From a (60) m tower, a car’s angles of depression change from (30^\circ) to (60^\circ):

d₁ = 60/tan 30° = 60√3 m
d₂ = 60/tan 60° = 20√3 m
Distance travelled = d₁ - d₂ = 40√3 m

If this occurs in (10) seconds, its speed is (4\sqrt3) m/s.

• Direction check: Determine whether the object remains on one side of the observer or passes to the opposite side before subtracting or adding distances.
• Assumption: Standard problems normally assume constant speed, level ground, and negligible observer dimensions.

III. Seating Arrangements — Positional Deduction

Seating arrangements require individuals or objects to be placed according to relational clues. A temporary diagram should be built from definite clues first, followed by relative and negative conditions.

A. Linear seating arrangement

A linear arrangement places people in one row or in parallel rows, with direction determining the meaning of left and right.

• Facing north: A person’s left matches the solver’s left, and the person’s right matches the solver’s right.
• Facing south: A person’s left appears on the solver’s right, and the person’s right appears on the solver’s left.
• Immediate neighbour: “A sits immediately left of B” means no person sits between A and B.
• Position gap: If A is third to the left of B, two seats lie between them.
• End positions: A row of (n) seats has positions (1) and (n) at its ends; the middle is ((n+1)/2) only when (n) is odd.
• Two-row arrangements: Opposite positions should be aligned vertically, and each row’s facing direction must be marked before interpreting lateral clues.
• Efficient construction:
  1. Place fixed positions such as “at an end” or “fourth from the left.”
  2. Insert linked blocks such as A–B or C–D–E.
  3. Apply negative clues such as “P is not adjacent to Q.”
  4. Test remaining alternatives against every clue.
• Worked example: Five people A, B, C, D, and E face north. C is in the middle, A is immediately left of C, E is at the right end, and B is left of A. The only arrangement is:

B  A  C  D  E
1  2  3  4  5

• Verification: A completed row is valid only if every clue holds simultaneously; satisfying most clues is insufficient.

B. Circular seating arrangement

A circular arrangement places participants around a circle, so relative direction matters but absolute rotation usually does not.

• Reference placement: Fix one person anywhere to remove rotationally equivalent arrangements.
• Facing centre: A person’s left is clockwise and right is anticlockwise.
• Facing outside: A person’s left is anticlockwise and right is clockwise.
• Immediate neighbour: Each person normally has exactly two immediate neighbours, one on each side.
• Opposite position: With an even number (n), the person opposite is (n/2) seats away in either direction.
• Clock-face wording: “At 3 o’clock” usually indicates the right side of the reference position, while “at 9 o’clock” indicates the left; the stated facing conditions still control personal left and right.
• Mixed-facing cases: Mark each participant with an inward or outward arrow before processing directional clues.
• Worked example: Six people face the centre. If B sits immediately left of A, place B one position clockwise from A. If D sits opposite A, place D three seats from A. Remaining clues then determine the unoccupied positions.
• Reflection issue: Clockwise and anticlockwise arrangements are not interchangeable when clues explicitly mention left or right.
• Final check: Read each clue from the named person’s viewpoint rather than from the solver’s viewpoint.

IV. Inequalities and Their Codes — Ordered Relationships

An inequality compares values without necessarily stating their exact magnitudes. Coded inequalities test whether symbolic statements can be decoded and combined through valid transitive reasoning.

A. Basic concepts of inequalities

Basic inequality analysis depends on recognizing signs, reversing relations correctly, and identifying conclusions supported by all statements.

• Standard relations:
  • (A>B): A is greater than B.
  • (A<B): A is less than B.
  • (A\ge B): A is greater than or equal to B.
  • (A\le B): A is less than or equal to B.
  • (A=B): A and B are equal.
  • (A\ne B): A and B are unequal.
• Transitivity: From (A>B) and (B>C), conclude (A>C).
• Compatible chain: From (A\ge B) and (B>C), conclude (A>C); at least one strict sign makes the final relation strict.
• Uncertain comparison: From (A>B) and (B<C), no definite relation between A and C follows.
• Reversal rule: Multiplying or dividing both sides by a negative number reverses the sign; for example, (x<4) implies (-x>-4).
• Coded signs: A question may define, for example, (A#B) as (A\ge B). Every code must first be replaced by its stated standard sign.
• Either–or conclusion: The pair (A>B) or (A\le B) is exhaustive, but such alternatives are used only when neither individual conclusion is independently certain.
• Worked example: Suppose (P@Q) means (P>Q), and (Q\%R) means (Q\ge R). Then:

P @ Q, Q % R
⇒ P > Q, Q ≥ R
⇒ P > R

• Constraint: Never infer the converse; (A>B) does not imply that a separately stated relationship caused or uniquely determined that order.

V. Algebraic Comparison — Roots as Ordered Values

Root-comparison problems provide two equations, usually quadratic, whose roots represent variables (x) and (y). The objective is to determine whether every possible pairing supports (x>y), (x<y), (x=y), or no definite relation.

A. Comparison of roots of equation

Comparison requires solving or locating both root sets and then comparing their complete ranges.

• Quadratic form:

ax² + bx + c = 0, where a ≠ 0

Here, (a), (b), and (c) are constants, while (x) is the unknown.

• Factorisation: Find numbers whose product is (ac) and sum is (b), split the middle term, and form linear factors.
• Quadratic formula:

x = (-b ± √(b² - 4ac))/(2a)

Here, (b^2-4ac) is the discriminant and (\pm) produces the two possible roots.

• Discriminant: If (D=b^2-4ac), then (D>0) gives two distinct real roots, (D=0) gives equal real roots, and (D<0) gives no real roots for ordinary magnitude comparison.
• Range method: If the smallest possible (x) exceeds the largest possible (y), then (x>y) for all pairings. Overlapping root ranges usually make the relation indeterminate.
• Worked example:

x² - 7x + 12 = 0  ⇒  (x - 3)(x - 4) = 0  ⇒  x = 3 or 4
y² - 3y + 2 = 0   ⇒  (y - 1)(y - 2) = 0   ⇒  y = 1 or 2

Since the minimum (x) is (3) and the maximum (y) is (2), (x>y) in every case.

• Indeterminate case: If (x\in{2,5}) and (y\in{3,4}), some pairings give (x<y) while others give (x>y); therefore, no unique comparison follows.
• Critical safeguard: Comparing only the larger roots or only one convenient pairing can produce a false conclusion; every possible root pairing must satisfy the claimed relation.