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Coordinate Geometry
Every CAT Coordinate Geometry formula on one page — distance, section formula, line equations, circles, and locus — each with a worked example.
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Trigonometry
CAT'26 QUANT CHEATSHEET
Every trigonometry formula and CAT shortcut you need for CAT 2026 — on one page.
Trigonometry on CAT rarely shows up as a pure identity question — it hides inside height-and-distance word problems, triangle-area shortcuts, and the occasional bound on an expression like a sinθ plus b cosθ. The habit that saves the most time is memorising the standard angle table cold and sketching the right triangle before touching any formula. This sheet lays out every formula you need for the CAT quant section: the basic ratios and identities, sum, difference and double-angle formulas, height and distance problems, the law of sines and cosines, and quadrant signs, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Trigonometry: every formula you need
1Basic Trig Ratios & Reciprocals
The six ratios defined from a right triangle's sides.
sinθ=opp/hyp, cosθ=adj/hyp, tanθ=opp/adj | cosec=1/sin, sec=1/cos, cot=1/tan
Example: right triangle 3-4-5, θ opposite the side 3 → sinθ=3/5, cosθ=4/5, tanθ=3/4.
CAT Hack: Reciprocal identities let you flip a known ratio instantly instead of recomputing from the triangle.
2Standard Angle Values
The exact sin, cos and tan values worth memorising outright.
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | — |
Example: sin30°+cos60° = 1/2+1/2 = 1.
CAT Favourite: Memorise this table cold — it saves real time versus deriving values mid-question.
3Pythagorean Identities
Three identities, all derived from the same right-triangle relation.
sin2θ+cos2θ=1 | 1+tan2θ=sec2θ | 1+cot2θ=cosec2θ
Example: tanθ=3/4 → sec2θ=1+9/16=25/16 → secθ=5/4.
4Complementary Angle Relations
Swapping sin and cos (etc.) when the angle is measured from 90°.
sin(90°−θ)=cosθ | cos(90°−θ)=sinθ | tan(90°−θ)=cotθ
Example: sin50° = cos40°.
5Sum & Difference Formulas
Expand the sine or cosine of a sum into two simpler terms.
sin(A±B)=sinAcosB±cosAsinB | cos(A±B)=cosAcosB∓sinAsinB
Example: sin75°=sin(45°+30°) → (√6+√2)/4.
CAT Hack: Build any non-standard angle (15°, 75°, 105°...) as a sum or difference of 30°, 45° and 60°.
6Double Angle Formulas
Express a function of 2θ purely in terms of θ.
sin2θ=2sinθcosθ | cos2θ=1−2sin2θ=2cos2θ−1
Example: θ=30° → sin60°=2sin30cos30=2(1/2)(√3/2) = √3/2.
7Range of a sinθ + b cosθ
This combination is always bounded by a single radical expression.
max = √(a2+b2), min = −√(a2+b2)
Example: 3sinθ+4cosθ → max=5, min = −5.
CAT Favourite: Instantly bounds any expression of this shape without calculus — a very common CAT shortcut.
8Angle of Elevation & Depression
The angle above or below the horizontal line of sight.
tanθ = height ÷ horizontal distance
Example: tower 100m tall, elevation angle 30° → distance = 100/tan30 = 100√3 ≈ 173.2 m.
CAT Insight: The angle of elevation from A to B always equals the angle of depression from B to A — they're alternate angles.
9Height & Distance: Two Points, Same Line
Find a height using two angles measured from points a known distance apart.
h = d ÷ (cotα − cotβ)
Example: angles 30° and 60° from points 20m apart → h = 20/(cot30−cot60) = 10√3 ≈ 17.3 m.
10Height & Distance: Depression to Two Objects
Find the distance between two objects seen from a known height.
distance = h ÷ (cotβ − cotα), α>β
Example: cliff 50m, depression angles 45° and 30° → 50(cot30−cot45) = ≈36.6 m.
Common Mistake: Always sketch the figure first — mixing up which angle belongs to which point is the most common slip here.
11Law of Sines
Relates every side of a triangle to the sine of its opposite angle.
a/sinA = b/sinB = c/sinC = 2R (R = circumradius)
Example: a=10, A=30°, B=45° → b = 10·sin45/sin30 = 10√2 ≈ 14.1.
12Law of Cosines
Finds a side (or angle) of any triangle, not just right triangles.
c2 = a2+b2−2ab·cosC
Example: a=5, b=7, C=60° → c2=25+49−35=39 → c ≈ 6.24.
CAT Hack: Use the law of cosines whenever you know two sides and the included angle, or all three sides.
13Area of a Triangle (Trig Form)
Compute area directly from two sides and the angle between them.
Area = ½ab·sinC
Example: a=6, b=8, C=30° → ½·6·8·0.5 = 12.
14Trig Equations: General Solutions
The full family of angles satisfying a basic trig equation.
sinθ=sinα → θ=nπ+(−1)nα | cosθ=cosα → θ=2nπ±α
Example: sinθ=1/2 → α=30° → θ=n·180°+(−1)ⁿ·30°.
15Periodicity & Quadrant Signs
Which ratios are positive in each quadrant, and how often they repeat.
ASTC: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4) are positive | sin, cos period 360°; tan period 180°
Example: sin200° is in Q3, where only tan is positive → sin200° is negative.
CAT Hack: Remember it as “All Students Take Calculus” — A, S, T, C by quadrant, in order.
CAT exam shortcuts, traps & revision
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CAT Exam Shortcuts
- Memorise the 0°/30°/45°/60°/90° value table cold
- Build non-standard angles as sums/differences of 30°, 45°, 60°
- max/min of a sinθ+b cosθ = ±√(a²+b²), no calculus needed
- Elevation from A to B = depression from B to A (alternate angles)
- Law of cosines whenever a triangle isn't right-angled and you know 2 sides + included angle
- ASTC rule: All, Sin, Tan, Cos positive by quadrant (Q1→Q4)
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Most Common CAT Traps
- Forgetting tan90° is undefined, not a large finite number.
- Picking the wrong sign for a ratio because the quadrant was misread.
- Mixing up which angle (elevation or depression) belongs to which observation point.
- Using cot as if it were tan, instead of its reciprocal.
- Applying the right-triangle-only formulas (like plain SOH-CAH-TOA) to a non-right triangle.
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30-Second Revision Box
- sin2θ+cos2θ=1; memorise standard angle values
- sin(A±B), cos(A±B); sin2θ=2sinθcosθ
- max/min of a sinθ+b cosθ = ±√(a²+b²)
- Elevation/depression: tanθ = height ÷ distance
- Law of sines: a/sinA = 2R; law of cosines: c²=a²+b²−2ab cosC
- ASTC quadrant rule; sin/cos period 360°, tan period 180°
Trigonometry rewards recognising the setup over memorising every identity separately — once a height-and-distance question is sketched correctly, or a non-right triangle is spotted as a law-of-cosines case, the answer is one substitution away. Drill this sheet until the standard angle table and the elevation-depression relationship are reflex, then test them on full sets and track progress with the CAT score predictor. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
