Mathematics

Miscellaneous — Trigonometry & Number Series

The miscellaneous section tests basic trigonometry (height-distance, identities) and number-series pattern recognition. These are easy 2–3 marks if you have the standard values memorised — and number series is a regular feature of every Banking, SSC and RRB Reasoning paper.

Exam relevance: SSC CGL gives 2-3 trigonometry questions in Tier-I and full theorem-based questions in Tier-II. Banking exams (IBPS, SBI, RBI) use number series in both Quant and Reasoning sections.

On this page

  1. Trigonometry — Basic Identities
  2. Heights & Distances
  3. Number Series

1Trigonometry — Basic Identities

Trigonometry studies the relations between sides and angles of a triangle.

Six ratios: sin = opp/hyp, cos = adj/hyp, tan = opp/adj, cosec = 1/sin, sec = 1/cos, cot = 1/tan. Standard angles (memorise the table 0°, 30°, 45°, 60°, 90°). Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. Complementary angles: sin(90°−θ) = cosθ, tan(90°−θ) = cotθ, sec(90°−θ) = cosecθ.

Examples
  • If sinθ = 3/5, find cosθ: cosθ = √(1−9/25) = 4/5.
  • Simplify (1 − sin²30°) = 1 − 1/4 = 3/4 = cos²30°.
  • If tanθ = 1, θ = 45°; sin45° = cos45° = 1/√2.
  • sin60°×cos30° + sin30°×cos60° = (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1 = sin90°.
Exam tip: Memorise the standard-angle table thoroughly. Most exam trigonometry questions are direct substitutions into these values.

2Heights & Distances

Heights & Distances is the application of trigonometry to find unknown heights or distances using angles of elevation and depression.

Angle of elevation = angle measured upward from the horizontal to an object above. Angle of depression = angle measured downward from the horizontal to an object below. Always draw a right-triangle and label the angle, base and height. tan(angle) = perpendicular/base is the most-used formula. For two angles of elevation from two distances, set up two equations and eliminate the height.

Examples
  • From a point 30 m from the base of a tower, the angle of elevation is 60°. Height = 30 × tan60° = 30√3 ≈ 51.96 m.
  • A man on a 50 m cliff sees a boat at angle of depression 30°. Distance = 50 × cot30° = 50√3 ≈ 86.6 m.
  • From the top of a 60 m building, the angle of depression of a car is 45° → car is 60 m from the base.
  • Elevation from two points at 30° and 60° on the same side, distance between points = 20 m → height = 10√3.
Exam tip: The angle of depression from A to B equals the angle of elevation from B to A — they are alternate interior angles.

3Number Series

A number series is a sequence of numbers arranged according to a definite rule. The task is to find the next term, the missing term, or the wrong term.

Common patterns: (i) Arithmetic progression — constant difference. (ii) Geometric progression — constant ratio. (iii) Squares/cubes — n², n³, or shifted versions. (iv) Mixed/double-difference series — differences themselves form a pattern. (v) Alternate series — odd and even terms follow separate rules. (vi) Prime numbers, Fibonacci-style series. Strategy: compute first differences; if not constant, compute second differences; check for ×, ÷, ², ³, +primes patterns.

Examples
  • AP: 5, 9, 13, 17, ?, ? → +4 each → 21, 25.
  • GP: 3, 6, 12, 24, ? → ×2 each → 48.
  • Squares: 1, 4, 9, 16, 25, ? → 36 (n²).
  • Mixed: 2, 6, 12, 20, 30, ? → differences 4,6,8,10,12 → next = 42.
Exam tip: When stuck, try ÷ between consecutive terms — if all divisions give equal ratios, it's a GP. If first differences are constant it is an AP. Otherwise look at squares or cubes.

Short Tricks & Shortcuts

Use these speed tricks in the exam. Each trick is followed by a worked example so you can verify the shortcut yourself.

Trick 1Standard-angle table — burn it into memory

Memorise sin & cos for 0°, 30°, 45°, 60°, 90°. sin: 0, 1/2, 1/√2, √3/2, 1. cos is the same values reversed. tan = sin/cos.

Example: sin60° × cos30° = (√3/2) × (√3/2) = 3/4. Done in one line.
Trick 2Convert everything to sin and cos

For any tricky identity, replace tan, cot, sec, cosec by their sin/cos equivalents and use sin²θ + cos²θ = 1. Most identities collapse in 2-3 steps.

Example: Prove (1 − cos²θ) = sin²θ — direct from the Pythagorean identity.
Trick 3Angle of depression = angle of elevation

The angle of depression from A down to B equals the angle of elevation from B up to A. Use whichever side gives an easier triangle.

Example: Cliff 50 m, depression 30° to a boat → distance = 50 × cot30° = 50√3 m.
Trick 4Number-series first-difference test

Take differences of consecutive terms. If constant → AP. If they form their own pattern (4, 6, 8, 10…) → mixed series. Else, try ratios for GP.

Example: 2, 6, 12, 20, 30, ? — diffs 4, 6, 8, 10, next diff 12 → answer 42.
Trick 5Sum of an AP / GP — quick formulas

AP sum = (n/2) × (first + last) or (n/2)[2a + (n−1)d]. GP sum = a(rⁿ − 1)/(r − 1) for r ≠ 1.

Example: Sum of 1+3+5+…+99 = (50/2)(1+99) = 2,500.
Trick 6Spotting the wrong term in a series

Compute first differences. The wrong term is the one whose difference breaks the otherwise regular pattern. Replace it with the value that fits.

Example: 5, 11, 17, 24, 29: diffs are 6, 6, 7, 5 — 24 breaks the +6 pattern; correct value is 23.

Quick Revision Facts

  • sin0° = 0, sin30° = 1/2, sin45° = 1/√2, sin60° = √3/2, sin90° = 1.
  • tan0° = 0, tan45° = 1, tan90° is undefined (infinity).
  • Sum of n terms of an AP = (n/2) × (first + last) or (n/2)[2a + (n−1)d].
  • Sum of n terms of a GP = a(rⁿ−1)/(r−1), for r ≠ 1.

Frequently Asked Questions

Convert everything to sin and cos using definitions, simplify, and use sin²θ + cos²θ = 1. Most government-exam identities collapse in 2-3 steps once everything is in sin/cos.

Compute the differences between consecutive terms. The wrong term will produce a difference that breaks the otherwise regular pattern. Replace it with the term that fits the pattern.
All Math TopicsExam Preparation Hub