Trigonometry

high importance~3 Q in Tier 119 formulas⚡ 11 shortcuts5 subtopics

Overview Learn (5)Formulas (19)Examples (5)Practice (72)

Ratios, the standard-value table, the three identities, complementary angles, value-putting and max-min values. Tier 1 reliably holds 2-4 trig questions and they fall almost mechanically to the value table, one identity, or a 3-4-5 style triangle — among the cheapest marks in the paper.

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Learn — 5 subtopics

One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.

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Formula sheet

Every formula on one printable page, grouped by subtopic.

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Solved examples

5 exam-level questions worked step by step.

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Practice & test

72 questions — untimed practice or a timed test with analysis.

Track record in the exam

avg 2.0 Q / shift2024: 2–3 Q2025: 2 Q

Questions per shift in recent SSC CGL papers.

Test difficulty mix (72 questions)

19 easy40 medium13 hard

Question patterns exams keep repeating

Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.

Standard values at special angles

very common

Spot it:

A sum/difference/product of sin/cos/tan (often sec/cosec/cot too) at 0°, 30°, 45°, 60°, 90° asked as one value.

How to solve: Convert every function to its table value via the $\frac{\sqrt{k}}{2}$ construction (reciprocals for sec/cosec/cot), then compute with exact fractions. Keep surds as surds; rationalise only the final answer.

Example: The value of $\tan 45^\circ + \cot 45^\circ$ is:

$\tan 45^\circ = 1$ and $\cot 45^\circ = 1$ : sum $= 2$ .

Learn this in “Ratios and standard values” →

One ratio given → the rest of the triangle

very common

Spot it:

'If sinθ = 3/5 find tanθ' style — one ratio (sometimes surd form), other ratios asked.

How to solve: Draw the right triangle from the given ratio, complete the third side by Pythagoras (expect a triplet), and read the asked ratio off the same triangle w.r.t. the named angle.

Example: If $\cosec\theta = \frac{13}{5}$ ( $\theta$ acute), the value of $\tan\theta$ is:

$\sin\theta = \frac{5}{13}$ → triangle 5-12-13: $\tan\theta = \frac{5}{12}$ .

Learn this in “Ratios and standard values” →

Conjugate pairs: sec±tan, cosec±cot

very common

Spot it:

'If secθ + tanθ = m, find secθ − tanθ' — or the pair subtracted/added to extract an individual ratio.

How to solve: $(\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1$ , so the conjugate is its reciprocal $\frac{1}{m}$ . Adding/subtracting the pair gives $2\sec\theta$ and $2\tan\theta$ individually; a triangle then returns sin/cos.

Example: If $\cosec\theta + \cot\theta = 3$ ( $\theta$ acute), the value of $\cosec\theta - \cot\theta$ is:

$\cosec\theta - \cot\theta = \frac{1}{\cosec\theta + \cot\theta} = \frac13$ .

Learn this in “Fundamental identities” →

sinθ + cosθ = m family

very common

Spot it:

A sum (or difference) of sin and cos given as m; the product sinθcosθ, the opposite difference, or a cubic combination asked.

How to solve: Square the given: $\sin\theta\cos\theta = \frac{m^2-1}{2}$ ; $(\sin\theta - \cos\theta)^2 = 2 - m^2$ ; cubes via $x^3+y^3=(x+y)(x^2-xy+y^2)$ . Sub-specials: $m = \sqrt2$ → product $\frac12$ ; $m = 1$ → product $0$ .

Example: If $\sin\theta + \cos\theta = \frac{6}{5}$ , the value of $\sin\theta\cos\theta$ is:

$\frac{m^2-1}{2} = \frac{\frac{36}{25}-1}{2} = \frac{11}{50}$ .

Learn this in “Value-putting and given-ratio questions” →

tanθ given → linear fraction in sin and cos

very common

Spot it:

tanθ = p/q (or 5tanθ = 4 style); a fraction like (a·sinθ + b·cosθ)/(c·cosθ + d·sinθ) asked.

How to solve: Divide numerator and denominator by $\cos\theta$ : the fraction becomes $\frac{a\tan\theta + b}{c + d\tan\theta}$ . Substitute $t = \frac{p}{q}$ exactly (fractions, not decimals) and simplify once.

Example: If $\tan\theta = \frac{4}{3}$ , the value of $\frac{3\sin\theta + 2\cos\theta}{3\cos\theta - 2\sin\theta}$ is:

Divide by $\cos\theta$ : $\frac{3\cdot\frac43 + 2}{3 - 2\cdot\frac43} = \frac{6}{\frac13} = 18$ .

Learn this in “Value-putting and given-ratio questions” →

Complementary pairs collapse a sum/product

very common

Spot it:

Angles pair to 90° (35 & 55, 20 & 70...) across different functions — or a long tangent chain with 45° in the middle.

How to solve: Flip one function of each pair: $\sec 55^\circ = \frac{1}{\sin 35^\circ}$ etc.; products like $\tan\theta\tan(90^\circ-\theta) = 1$ . Chains $\tan 1^\circ \cdots \tan 89^\circ$ collapse to 1. Difference-of-twins questions ( $\cos 50^\circ - \sin 40^\circ$ ) are instantly 0.

Example: The value of $\tan 10^\circ \tan 20^\circ \tan 70^\circ \tan 80^\circ$ is:

Pairs: $10+80 = 90$ and $20+70 = 90$ ; each pair multiplies to 1, so the product is 1.

Learn this in “Complementary angles” →

Angle from a complementary-function equality

very common

Spot it:

'If sin 3A = cos(A − 10°), find A' — different functions at related angles, one unknown.

How to solve: Force the same function ( $\cos x = \sin(90^\circ - x)$ ) or use the shortcut: complementary-function arguments sum to $90^\circ$ . Solve the resulting linear equation; sanity-check the angle keeps all arguments in range.

Example: If $\tan 2\theta = \cot(\theta - 12^\circ)$ , the value of $\theta$ is:

$2\theta + \theta - 12^\circ = 90^\circ \Rightarrow 3\theta = 102^\circ \Rightarrow \theta = 34^\circ$ .

Learn this in “Complementary angles” →

Max/min of a·sinθ ± b·cosθ

very common

Spot it:

'Find the maximum of 4sinθ + 3cosθ' — Pythagorean coefficients; minimum may be asked instead.

How to solve: Max $= +\sqrt{a^2+b^2}$ , min $= -\sqrt{a^2+b^2}$ (unrestricted θ); with $0^\circ \le \theta \le 90^\circ$ check endpoints too. Expect triplets: (3,4,5), (5,12,13), (8,15,17).

Example: The maximum value of $8\sin\theta - 15\cos\theta$ is:

$\sqrt{8^2 + 15^2} = 17$ (minimum $-17$ ).

Learn this in “Maximum and minimum values” →

AM-GM bounds and sinθcosθ ≤ ½

common

Spot it:

Minimum of tan²+cot², sec²+cosec², tan+cot — or the max of sinθcosθ / 2sinθcosθ.

How to solve: $x + \frac1x \ge 2$ (equality $x=1$ , i.e. 45°); $\sec^2\theta + \cosec^2\theta = \frac{1}{\sin^2\theta\cos^2\theta} \ge 4$ ; $\sin\theta\cos\theta \le \frac12$ peaking at 45°.

Example: The minimum value of $\tan\theta + \cot\theta$ ( $\theta$ acute) is:

$\tan\theta + \frac{1}{\tan\theta} \ge 2$ , equality at $\theta = 45^\circ$ .

Learn this in “Maximum and minimum values” →

Two sides of the right triangle → a named ratio

common

Spot it:

Two sides given (or hyp + one leg); sin/cos/tan of one acute angle asked — the angle described in words ('opposite the longer leg').

How to solve: Third side by Pythagoras, then attach opposite/adjacent to the NAMED angle — 'opposite the shorter leg' is the smaller angle. Triplet multiples supply the exam numbers.

Example: In a right triangle, the legs adjacent to and opposite an acute angle are $15$ cm and $8$ cm. The hypotenuse is:

$\sqrt{15^2 + 8^2} = \sqrt{289} = 17$ cm (8-15-17).

Learn this in “Ratios and standard values” →

Your next step

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