Trigonometry Questions for CDS

Standard Angle Values

CDS directly asks for values like sin 75, cos 15, tan 22.5, or asks you to evaluate an expression. You must know the table for 0, 30, 45, 60, 90, and derived values for 120, 135, 150, 180, 270, 360. The pattern is simple: for sine, the values go root 0, root 1, root 2, root 3, root 4 — all divided by 2.

• sin 0=0, sin 30=1/2, sin 45=1/root2, sin 60=root3/2, sin 90=1
• cos is the reverse of sin: cos 0=1, cos 30=root3/2, cos 45=1/root2, cos 60=1/2, cos 90=0
• tan = sin/cos: tan 30=1/root3, tan 45=1, tan 60=root3, tan 90=undefined
• For angles beyond 90: use CAST rule — All positive in Q1, Sin in Q2, Tan in Q3, Cos in Q4
• sin(180-x)=sin x, cos(180-x)=-cos x, tan(180+x)=tan x — these are very frequently tested
• sin(-x)=-sin x (odd function), cos(-x)=cos x (even function)

Fundamental Trigonometric Identities

These identities are the backbone of most simplification questions in CDS. You will be given an expression and asked to simplify it to a number or simpler form. The three Pythagorean identities are the most tested. Learn them in all three rearranged forms each.

• sin^2 x + cos^2 x = 1 (rearranged: sin^2 x = 1 - cos^2 x, cos^2 x = 1 - sin^2 x)
• 1 + tan^2 x = sec^2 x (rearranged: sec^2 x - tan^2 x = 1, tan^2 x = sec^2 x - 1)
• 1 + cot^2 x = cosec^2 x (rearranged: cosec^2 x - cot^2 x = 1)
• tan x = sin x / cos x, cot x = cos x / sin x — use these to convert everything to sin and cos
• sec x = 1/cos x, cosec x = 1/sin x — reciprocal relations are tested in simplification
• If given sec^2 x - tan^2 x, the answer is always 1 — a favorite CDS direct question

Compound and Multiple Angle Formulas

CDS regularly asks for values like sin 75, sin 15, cos 75, tan 15 and occasionally double angle results. The compound angle formulas let you break these into standard angles. Also, double angle formulas are used in simplification — especially sin 2x and cos 2x in their three forms.

• sin(A+B) = sinA cosB + cosA sinB
• sin(A-B) = sinA cosB - cosA sinB
• cos(A+B) = cosA cosB - sinA sinB
• cos(A-B) = cosA cosB + sinA sinB
• sin 2x = 2 sinx cosx, cos 2x = cos^2 x - sin^2 x = 1 - 2sin^2 x = 2cos^2 x - 1
• tan(A+B) = (tanA + tanB) / (1 - tanA tanB)

Heights and Distances

This is the real-world application section. CDS asks 2 to 3 word problems every paper. You are given a scenario — tower, shadow, angle of elevation or depression — and must set up a right triangle and solve. Speed comes from recognizing the standard tan-based formula instantly. Always draw a rough figure before solving.

• Angle of elevation: looking UP at an object from ground level
• Angle of depression: looking DOWN at an object from a height — use alternate angles property to convert
• tan(angle) = opposite/adjacent = height/horizontal distance — this is the main formula
• For two-angle problems (observer moves), set up two equations in two unknowns
• Common setup: tower height h, distance d, angle theta — h = d * tan(theta)
• When angle is 45, tan=1, so height = distance. When angle is 60, height = distance * root3

Complementary Angle and Co-function Relations

CDS loves asking simplification questions that collapse using co-function identities. If two angles add to 90, their trig functions swap. This is used to simplify long expressions rapidly. Also, product identities like sin(90-x) = cos x appear in series-type simplification questions.

• sin(90-x) = cos x, cos(90-x) = sin x
• tan(90-x) = cot x, cot(90-x) = tan x
• sec(90-x) = cosec x, cosec(90-x) = sec x
• Used to collapse paired terms: sin 20 * cosec 70 = sin 20 * cosec(90-20) = sin 20 * sec 20 — simplifies quickly
• In long series questions: sin 1 * sin 89 + sin 2 * sin 88 + ... — use pairing strategy

• Confusing sin(A+B) with sinA + sinB — there is no such identity. This wrong expansion kills marks in simplification questions.
• Forgetting the sign in cos(A+B) = cosA cosB MINUS sinA sinB — students often put a plus sign and get wrong answers.
• In heights and distances, using sine instead of tangent — if the hypotenuse is not given, you need tan, not sin.
• Not applying the CAST rule and giving positive values for angles in Q2, Q3, Q4 — always check the quadrant first.
• In double angle, writing cos 2x = cos^2 x + sin^2 x (which is just 1) instead of cos^2 x - sin^2 x — sign error is the classic trap.

• Sine table trick: sin 0, 30, 45, 60, 90 = root(0/4), root(1/4), root(2/4), root(3/4), root(4/4). Then cos is the reverse.
• CAST rule — go anticlockwise from Q1: All Students Take Calculus (All, Sin, Tan, Cos positive in Q1, Q2, Q3, Q4).
• For compound angle cos: C-O-S = Cos-Cos (multiply) then OPPOSITE sign for Sin-Sin. For sin: SIN-COS + COS-SIN (same sign for both add and subtract cases, just flip middle sign).
• Heights and distances: think TOH CAH SOA backwards — for height problems Tan = Opposite over Adjacent is your default weapon.

• CDS typically gives 5 to 7 trigonometry questions. Expect 2 direct identity simplification, 1 to 2 standard angle value questions, and 2 heights and distances word problems per paper.
• Identities questions often have the form: if sin x + cos x = k, find sin x * cos x or sin^2 x - cos^2 x. These are solved by squaring the given equation — a very repeatable pattern.
• Heights and distances problems in CDS use angles 30, 45, 60 almost exclusively. You will never need a table or calculator — the numbers always work out cleanly.
• Time allocation: spend no more than 90 seconds per identity question and 2 minutes per height-distance problem. If you are stuck beyond that, mark and move on.
• Do not ignore the cosec, sec, cot questions — CDS papers frequently test sec^2 - tan^2 = 1 and cosec^2 - cot^2 = 1 as one-step direct questions worth free marks.