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SSC CGL Basic Trig Ratios

Study Material — 3 PYQs (2018–2020) · Concept Notes · Shortcuts

SSC CGL Basic Trig Ratios is a frequently tested subtopic — 3 previous year questions from 2018–2020 papers are included below with concept notes, key rules and shortcut tricks.

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2018–2020
27 Practice
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10 Key Points
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Previous Year Questions

SSC CGL Basic Trig Ratios — Past Exam Questions

3 questions from actual SSC CGL papers · all shown free · click option to reveal solution

Exam Q 12018Previous Year Pattern

If sin θ = 5/13, find the value of (tan θ + cos θ).

Exam Q 22019Previous Year Pattern

If sin θ + cos θ = 7/5, where θ is an acute angle, then find the value of tan θ + cot θ.

Exam Q 32020Previous Year Pattern

If sin θ + cos θ = 7/5, where θ is an acute angle, then find the value of tan θ + cot θ.

Concept Notes

Basic Trig Ratios— Rules & Concept

Core ConceptRead this first — the foundation of the topic

BASIC TRIGONOMETRIC RATIOS — SSC CGL ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

CORE CONCEPT ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Trigonometry is the study of the relationship between angles and sides of a right-angled triangle. In a right triangle, there are three sides: Hypotenuse (longest side, opposite the 90° angle), Perpendicular (side opposite the given angle), and Base (side adjacent to the given angle). The six trig ratios are simply fractions made from these three sides. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Formula BlockMemorise — at least one formula appears in every paper

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

For angle θ in a right triangle:

sin θ = Perpendicular / Hypotenuse
cos θ = Base / Hypotenuse
tan θ = Perpendicular / Base
cosec θ = Hypotenuse / Perpendicular = 1/sin θ
sec θ = Hypotenuse / Base = 1/cos θ
cot θ = Base / Perpendicular = 1/tan θ

Memory Trick — SOH CAH TOA:

S = Sin = Opposite/Hypotenuse
C = Cos = Adjacent/Hypotenuse
T = Tan = Opposite/Adjacent

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

STANDARD ANGLE VALUE TABLE

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Angle: 0° 30° 45° 60° 90°

sin: 0 1/2 1/√2 √3/2 1

cos: 1 √3/2 1/√2 1/2 0

tan: 0 1/√3 1 √3 ∞

cosec: ∞ 2 √2 2/√3 1

sec: 1 2/√3 √2 2 ∞

cot: ∞ √3 1 1/√3 0

Shortcut to memorise sin values 0° to 90°:

Write 0, 1, 2, 3, 4 → divide by 4 → take square root.
sin 0° = √(0/4) = 0
sin 30° = √(1/4) = 1/2
sin 45° = √(2/4) = 1/√2
sin 60° = √(3/4) = √3/2
sin 90° = √(4/4) = 1

For cos: read this table in REVERSE order.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Key RulesCore rules you must know cold

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1. sin θ × cosec θ = 1 2. cos θ × sec θ = 1 3. tan θ × cot θ = 1 4. tan θ = sin θ / cos θ 5. cot θ = cos θ / sin θ 6. sin²θ + cos²θ = 1 (Most important identity) 7. 1 + tan²θ = sec²θ 8. 1 + cot²θ = cosec²θ From identity 6 you can also write: sin²θ = 1 - cos²θ and cos²θ = 1 - sin²θ From identity 7: sec²θ - tan²θ = 1 From identity 8: cosec²θ - cot²θ = 1 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Exam PatternsWhat examiners ask — read before attempting PYQs

— WHAT GETS ASKED ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ SSC CGL asks trig in 3 main ways: 1. Direct value substitution (put angle, calculate expression) 2. Identity-based simplification (simplify using sin²+cos²=1 type) 3. Given one ratio, find others (if sin θ = 3/5, find tan θ) Type 3 is very common.

Use the Pythagorean triplet shortcut here. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ SHORTCUT: PYTHAGOREAN TRIPLETS ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Common triplets to memorise: (3,4,5), (5,12,13), (8,15,17), (7,24,25) If sin θ = 3/5, then Perpendicular=3, Hypotenuse=5, so Base=4 Now you can find ALL six ratios instantly without any calculation. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Worked ExampleSolve this step-by-step before moving on
1
Step 1

sin θ = 5/13 → Perpendicular = 5, Hypotenuse = 13

2
Step 2

Use triplet (5, 12, 13) → Base = 12

3
Step 3

tan θ = P/B = 5/12

4
Step 4

cos θ = B/H = 12/13

5
Step 5

tan θ + cos θ = 5/12 + 12/13 = (5×13 + 12×12) / (12×13) = (65 + 144) / 156 = 209/156 Answer: 209/156 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ WORKED EXAMPLE 2 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Q: Find the value of: 2sin²30° + 3cos²45° - tan²60°

1
Step 1

sin 30° = 1/2, so sin²30° = 1/4

2
Step 2

cos 45° = 1/√2, so cos²45° = 1/2

3
Step 3

tan 60° = √3, so tan²60° = 3

4
Step 4

Put values: = 2(1/4) + 3(1/2) - 3 = 1/2 + 3/2 - 3 = 4/2 - 3 = 2 - 3 = -1 Answer: -1 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Exam TrapsCommon mistakes students make — avoid these

— THE #1 TRAP ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Students confuse sin²θ with sin(2θ). They are NOT the same. sin²θ = (sin θ)² — you square the value of sin θ sin(2θ) = 2 sin θ cos θ — it is a different angle formula Also, many students write tan 90° = 0. It is UNDEFINED (infinity).

This is a direct trap in MCQs. Never write a finite value for tan 90° or cosec 0°.

Key Points to Remember

  • sin θ = Perpendicular/Hypotenuse; cos θ = Base/Hypotenuse; tan θ = Perpendicular/Base
  • Memory trick SOH-CAH-TOA: Sin=Opposite/Hyp, Cos=Adjacent/Hyp, Tan=Opposite/Adjacent
  • Formula: sin²θ + cos²θ = 1 — the most used identity in SSC exams
  • Formula: 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ
  • Shortcut: sin values for 0°,30°,45°,60°,90° = √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, 1/√2, √3/2, 1
  • cos values are the REVERSE of sin values: cos 0°=1, cos 30°=√3/2, cos 60°=1/2, cos 90°=0
  • Shortcut: Pythagorean triplets (3,4,5), (5,12,13), (8,15,17), (7,24,25) solve ratio problems instantly
  • tan 45° = 1; sin 30° = cos 60° = 1/2; sin 60° = cos 30° = √3/2 — exam favourites
  • tan 90° and cosec 0° are UNDEFINED — never assign them a zero or finite value
  • Reciprocal pairs: sin↔cosec, cos↔sec, tan↔cot — their products always equal 1

Exam-Specific Tips

  • sin²θ + cos²θ = 1 is the Pythagorean trigonometric identity and holds true for ALL values of θ
  • tan 45° = 1, which means at 45° the perpendicular and base of a right triangle are always equal
  • sec²θ - tan²θ = 1 and cosec²θ - cot²θ = 1 — these two derived identities appear directly in simplification questions
  • The value of sin θ and cos θ always lies between -1 and 1 inclusive; they can never exceed 1
  • tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = undefined (not defined)
  • sin 30° = cos 60° = 1/2 — complementary angle rule: sin θ = cos(90°-θ) for all θ
  • The Pythagorean triplet (3, 4, 5) is the smallest and most frequently used triplet in SSC trig problems
  • cosec 30° = 2 and sec 60° = 2 — both equal 2, a common distractor in value-based MCQs
Practice MCQs

Basic Trig Ratios — Practice Questions

27graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 27

All MCQs →
Practice 1easy

If cos θ = 8/17 and θ is acute, what is the value of tan θ?

Practice 2easy

If sin 30° = 1/2, what is the value of cos 60°?

Practice 3easy

In a right-angled triangle, if sin A = 4/5, what is the value of tan A?

Practice 4easy

If cot θ = 7/24 and θ is in the first quadrant, find sin θ.

Practice 5easy

In a right-angled triangle, if one acute angle is 30°, what is the value of sin 30°?

Practice 6easy

In a right-angled triangle, the hypotenuse is 13 cm and one side is 5 cm. If θ is the angle opposite the side of length 5 cm, what is sin θ?

Practice 7easy

If tan 45° = 1, what is the value of cot 45°?

Practice 8medium

In a right-angled triangle, if tan A = 5/12, find the value of sec A.

Practice 9medium

If sin θ = cos θ, where 0° < θ < 90°, find the value of (sin³θ + cos³θ + 2 sin θ cos θ).

Practice 10medium

If tan A = 1 and tan B = 1/√3, where A and B are acute angles, find the value of (A + B) in degrees.

Practice 11medium

If sec θ - tan θ = 1/3, find the value of sec θ + tan θ.

Practice 12medium

In a right-angled triangle, if tan α = 7/24, find the value of (sin α + cos α) / (sin α - cos α).

Practice 13medium

If sec θ - tan θ = 2/5, find the value of sec θ + tan θ.

Practice 14medium

A ladder leans against a wall making an angle of 60° with the ground. If the ladder is 10 metres long, how far is the base of the ladder from the wall?

Practice 15medium

If sin θ + cos θ = √2, find the value of sin θ · cos θ.

Practice 16medium

In a right triangle, if cot θ = 8/15, find the value of cosec θ.

Practice 17medium

If tan θ = 5/12 and θ is in the first quadrant, find the value of (13 sin θ - 5 cos θ).

Practice 18hard

If sin θ + cos θ = √2, find the value of sin θ cos θ.

Practice 19hard

If 3 sin θ + 4 cos θ = 5, find the value of 3 cos θ − 4 sin θ.

Practice 20hard

If tan α + cot α = 4, where 0° < α < 90°, find the value of tan² α + cot² α.

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60-Second Revision — Basic Trig Ratios

  • Remember: SOH-CAH-TOA — Sin=P/H, Cos=B/H, Tan=P/B; write this first on rough sheet
  • Formula: sin²θ + cos²θ = 1; sec²θ - tan²θ = 1; cosec²θ - cot²θ = 1 — memorise all three
  • Shortcut: For sin table, write √0/2 to √4/2; for cos, reverse the same sequence
  • Shortcut: If one ratio is given, use Pythagorean triplets (3-4-5, 5-12-13, 8-15-17) to find all others instantly
  • Trap: tan 90° is UNDEFINED not 0; cosec 0° is UNDEFINED not 1 — watch for these in MCQ options
  • Remember: sin θ = cos(90°-θ) and tan θ = cot(90°-θ) — complementary angle rule saves time in simplification
  • Trap: sin²θ and sin(2θ) are completely different — squaring the ratio is NOT the same as doubling the angle
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