Trigonometry Questions for SSC CGL

Basic Trigonometric Ratios

Trigonometry starts with a right-angled triangle. For a given angle A, the three sides are Opposite (perpendicular), Adjacent (base), and Hypotenuse. The six ratios — sin, cos, tan, cosec, sec, cot — are just combinations of these three sides. If you know sin and cos, everything else is derived. Think of it like knowing two ingredients to cook all six dishes.

• sin A = Opposite / Hypotenuse (O/H)
• cos A = Adjacent / Hypotenuse (A/H)
• tan A = Opposite / Adjacent (O/A) = sin A / cos A
• cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A
• These are reciprocal pairs: sin-cosec, cos-sec, tan-cot
• tan A x cot A = 1 always — use this to cancel terms fast

Standard Angle Values Table

SSC CGL directly substitutes standard angles (0, 30, 45, 60, 90 degrees) into expressions. You must recall these instantly without deriving them. The pattern for sin values from 0 to 90 degrees is root(0)/2, root(1)/2, root(2)/2, root(3)/2, root(4)/2 — that is 0, 1/2, 1/root2, root3/2, 1. Cos is the reverse of sin. Tan = sin/cos.

• sin 0=0, sin 30=1/2, sin 45=1/root2, sin 60=root3/2, sin 90=1
• cos 0=1, cos 30=root3/2, cos 45=1/root2, cos 60=1/2, cos 90=0
• tan 0=0, tan 30=1/root3, tan 45=1, tan 60=root3, tan 90=undefined
• sin(90-A)=cos A and cos(90-A)=sin A — the complementary angle rule
• For angles above 90 deg: sin 120=sin 60=root3/2, cos 120= -cos 60= -1/2

Fundamental Identities

Three Pythagorean identities are the backbone of almost every simplification question. SSC CGL loves giving an expression that looks complex but collapses to 1 or 0 if you apply the right identity. Treat them like algebraic substitution — replace a messy cluster with a clean number. The key skill is recognizing which identity to apply.

• Identity 1: sin2A + cos2A = 1 (most used)
• Identity 2: 1 + tan2A = sec2A, so sec2A - tan2A = 1
• Identity 3: 1 + cot2A = cosec2A, so cosec2A - cot2A = 1
• Derived: sin2A - 1 = -cos2A, tan2A - sec2A = -1 (sign flip versions)
• a2 - b2 = (a+b)(a-b) works on trig too: (secA+tanA)(secA-tanA) = 1

Height and Distance

These are word problems using tan, sin, cos in real-world scenarios — tower heights, shadow lengths, angle of elevation and depression. Angle of elevation is looking UP at an object. Angle of depression is looking DOWN. In both cases, the angle is always measured from the horizontal. The most tested angles are 30, 45, and 60 degrees. Draw a rough triangle first — it saves time and prevents sign errors.

• Angle of elevation and depression are alternate interior angles — they are equal
• tan(angle) = height / horizontal distance — the formula you use 90% of the time
• If two towers or two observers: set up two separate triangles sharing the base
• Tower height h, shadow length d: tan(elevation angle) = h/d
• When angle changes from 30 to 60, the horizontal distance ratio is root3 : 1

Compound and Supplementary Angle Shortcuts

SSC CGL occasionally asks simplification using compound angle formulas or allied angle rules. You do not need to derive them but must recognize patterns quickly. The allied angle rules for 180-A, 180+A, 360-A reduce to basic ratios and save time in MCQs.

• sin(180-A) = sin A, cos(180-A) = -cos A
• sin(180+A) = -sin A, cos(180+A) = -cos A
• sin(360-A) = -sin A, cos(360-A) = cos A
• sin(A+B) = sinA cosB + cosA sinB
• cos(A+B) = cosA cosB - sinA sinB
• tan(A+B) = (tanA + tanB) / (1 - tanA tanB)

• Confusing angle of elevation with angle of depression: both are measured from horizontal, not vertical. Draw a diagram every time.
• Writing sin2A as (sinA)2 is correct but forgetting that sin(2A) is a completely different thing — double angle formula is 2.sinA.cosA.
• Flipping sec2A - tan2A = 1 as tan2A - sec2A = 1: the answer becomes -1, which changes everything in simplification.
• Using tan = opposite/adjacent but accidentally flipping the triangle sides when the angle is at the top of the triangle instead of the base.
• Forgetting that tan 90 is undefined — if an expression simplifies to tan 90, the answer is undefined or the question has an error, not zero.

• SOH-CAH-TOA: Sin=Opposite/Hypotenuse, Cos=Adjacent/Hypotenuse, Tan=Opposite/Adjacent. Say it like a chant.
• For the sin value table 0 to 90: think 0,1,2,3,4 under a root divided by 2: root0/2=0, root1/2=1/2, root2/2=1/root2, root3/2, root4/2=1. Cos is just reverse.
• ASTC (All Silver Tea Cups) for signs in 4 quadrants: Q1=All positive, Q2=Sin positive, Q3=Tan positive, Q4=Cos positive.
• For identity recall: SEC-TAN pair and COSEC-COT pair always equal 1 when multiplied as (x+y)(x-y). The middle identity sin2+cos2=1 is always the anchor.

• SSC CGL typically places 3 to 5 trig questions: 1 from identities simplification, 1 from standard angles substitution, 1 to 2 from height and distance, and sometimes 1 mixed expression. Identities and standard angles are the fastest marks — solve those first.
• For simplification questions, the answer is almost always a clean number like 0, 1, 2, or a simple fraction. If your answer is messy, you made an error — re-check identity application.
• Height and distance questions in Tier 1 use only 30, 45, 60 degree angles. Do not waste time with other angles unless Tier 2.
• When an expression has (secA - tanA) or (cosecA - cotA), immediately multiply and divide by its conjugate or use the product-equals-1 shortcut. This saves 90 seconds per question.
• In Tier 2, compound angle and double angle formulas appear. Tier 1 stays at identity and H-and-D level. Calibrate your preparation depth accordingly.