Trigonometry is one of the highest-scoring areas in the NDA written exam, with 25–30 marks tied to it almost every year. This Cavalier guide takes you from the six basic ratios to the big identity families, signs in the four quadrants, allied angles and equation solving. Master these and you can answer most trig questions in seconds.
Why Trigonometry Rules NDA Maths
Trigonometry literally means “measurement of triangles”. In the NDA paper it appears in two big ways: as direct ratio and identity questions, and as a tool inside height & distance, properties of triangles, calculus and even 3D geometry. So your marks here have a multiplier effect across the whole paper.
The good news for a busy Class 11–12 student is that this chapter is formula-driven and predictable. Once a handful of definitions and identities are memorised cold, the questions become almost mechanical. There is very little “trick” involved — the same ten or twelve formulas keep reappearing in slightly disguised forms year after year.
At The Cavalier we tell every cadet aspirant to treat trigonometry as a guaranteed scoring bank. Unlike calculus, where a single sign slip can wreck a long solution, most trig questions are short two-to-three-step problems. If your fundamentals are clean, you will rarely spend more than a minute on each one, leaving precious time for tougher sections.
Roughly 6–9 questions in NDA Maths come directly from trigonometric ratios, identities and equations. That is enough to move you across a cut-off, so do not skip it.
Measuring Angles: Degrees and Radians
An angle can be measured in degrees or radians. One full rotation is 360° or 2π radians. The radian is the “natural” unit used in calculus, so you must convert comfortably both ways. A radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius.
180° = π radians
1 radian = (180 ÷ π)° ≈ 57.3°
1° = (π ÷ 180) radians
Arc length: ℓ = rθ (with θ in radians)
So to change degrees to radians, multiply by π/180. To change radians to degrees, multiply by 180/π. For example, 30° = 30 × π/180 = π/6 radians, and π/3 radians = 60°. Keep the common pairs ready: 45° = π/4, 90° = π/2, 120° = 2π/3 and 270° = 3π/2.
The arc-length relation ℓ = rθ is itself worth a mark or two. If a wheel of radius r turns through angle θ, the distance a point on the rim travels is exactly rθ — provided θ is in radians. Forgetting that condition is the single most common slip here.
The grade (or gradian) system, where a right angle = 100 grades, occasionally appears. Relation: 90° = 100g = π/2 radians.
The Six Trigonometric Ratios
Take a right-angled triangle. For an acute angle θ, label the side opposite θ as perpendicular (P), the side touching θ (not the hypotenuse) as base (B), and the longest side as hypotenuse (H).
sinθ = P/H cosθ = B/H tanθ = P/B
cosecθ = H/P secθ = H/B cotθ = B/P
The classic memory aid is “Pandit Badri Prasad Har Har Bole” for P/H, B/H, P/B in the order sin, cos, tan.
Reciprocal links
- cosecθ = 1/sinθ
- secθ = 1/cosθ
- cotθ = 1/tanθ
Quotient links
- tanθ = sinθ/cosθ
- cotθ = cosθ/sinθ
cosec is the reciprocal of sin, not of cos. Many students wrongly pair sec with sin. Lock it in: co-secant goes with sine.
Standard-Angle Table You Must Memorise
The values for 0°, 30°, 45°, 60° and 90° appear in nearly every trig question. Memorise sin and cos; everything else follows.
sin: 0°→0, 30°→1/2, 45°→1/√2, 60°→√3/2, 90°→1
cos: 0°→1, 30°→√3/2, 45°→1/√2, 60°→1/2, 90°→0
tan: 0°→0, 30°→1/√3, 45°→1, 60°→√3, 90°→∞ (undefined)
A neat trick: write 0, 1, 2, 3, 4 under 0°, 30°, 45°, 60°, 90°. Take each number, divide by 4, then take the square root. That gives the sine values. Reverse the order for cosine.
tan90° and cot0° are undefined because you would be dividing by zero. Whenever a denominator hits cos90° or sin0°, expect “not defined”.
The Three Pythagorean Identities
These come straight from the Pythagoras theorem and are the backbone of every simplification question.
sin2θ + cos2θ = 1
1 + tan2θ = sec2θ
1 + cot2θ = cosec2θ
From the first identity you can derive useful rearrangements such as sin2θ = 1 − cos2θ and (1 − cosθ)(1 + cosθ) = sin2θ. The second and third are just the first divided through by cos2θ and sin2θ respectively, which is exactly how you should reconstruct them if your memory blanks in the hall.
Whenever an expression mixes secant with tangent, or cosecant with cotangent, the “difference” identities are usually the key. A favourite NDA setup gives you secθ − tanθ and asks for secθ + tanθ; since their product is (sec2θ − tan2θ) = 1, the second quantity is simply the reciprocal of the first.
sec2θ − tan2θ = 1 and cosec2θ − cot2θ = 1. These “difference” forms are what examiners love to disguise.
Signs in the Four Quadrants
As θ sweeps from 0° to 360°, the ratios change sign. The rule is captured by “All Silver Tea Cups” (also called the ASTC or CAST rule), reading anticlockwise from the first quadrant.
- Quadrant I (0°–90°): All ratios positive.
- Quadrant II (90°–180°): Only Sin (and cosec) positive.
- Quadrant III (180°–270°): Only Tan (and cot) positive.
- Quadrant IV (270°–360°): Only Cos (and sec) positive.
Students forget that cosec follows sin, and sec follows cos. If sin is positive in a quadrant, so is cosec; if cos is negative, so is sec.
So sin150° is positive (Quadrant II) while cos150° is negative. This single rule saves you from dozens of sign errors.
Allied Angles and Reduction Formulae
Allied angles are angles of the form (90° ± θ), (180° ± θ), (270° ± θ) and (360° ± θ). A simple two-step rule converts any of them to a first-quadrant ratio.
Step 1 (function change): If the angle is 90° or 270° ± θ (odd multiple of 90°), the ratio changes — sin↔cos, tan↔cot, sec↔cosec. If it is 180° or 360° ± θ (even multiple), the ratio stays the same.
Step 2 (sign): Decide the sign from the quadrant of the original angle using ASTC.
For example, sin(180° − θ) = sinθ (function unchanged, Quadrant II so sine positive). And cos(90° + θ) = −sinθ (function changed to sine, Quadrant II so cosine is negative there). Similarly tan(270° − θ) = cotθ, because 270° is an odd multiple of 90° so tan becomes cot, and 270° − θ sits in Quadrant III where tangent is positive.
Once this two-step routine becomes automatic, you can evaluate the trig ratio of any angle, however large or negative, without a calculator. That is precisely the skill examiners are testing when they throw a value like sin(1110°) at you — you reduce it within one full turn first, then apply the rule.
Negative angles: sin(−θ) = −sinθ, cos(−θ) = cosθ, tan(−θ) = −tanθ. Cosine is the “even” one; the rest are “odd”.
Compound and Multiple Angle Formulae
These extend trigonometry to sums, differences and multiples of angles. They turn up in identity proofs and equation questions.
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB −/+ sinA sinB
tan(A ± B) = (tanA ± tanB) ÷ (1 −/+ tanA tanB)
Double-angle forms
- sin2A = 2 sinA cosA
- cos2A = cos2A − sin2A = 1 − 2sin2A = 2cos2A − 1
- tan2A = 2tanA ÷ (1 − tan2A)
The three forms of cos2A let you switch between sin, cos and a constant — extremely handy when an equation mixes sin2 and cos2 terms.
Worked Example: Simplifying an Expression
Let us put the identities to work on a typical NDA-style simplification.
If sinθ = 3/5 and θ lies in the first quadrant, find the value of (secθ + tanθ).
So the answer is 2. Notice how the single fact sinθ = 3/5 unlocked every other ratio through the Pythagorean identity and the quotient rule. This 3-4-5 right triangle is one of three “Pythagorean triples” (alongside 5-12-13 and 8-15-17) that NDA loves, because the surds vanish and the arithmetic stays clean. Recognising them on sight saves real seconds.
A small but vital point: the question said θ is in the first quadrant, so we took the positive root for cosθ. Had it been in the second quadrant, cosθ would have been −4/5, and the final answer would change. Always read the quadrant condition before choosing a sign.
Solving Trigonometric Equations
A trigonometric equation is true only for certain values of the angle. The complete set of solutions is the general solution, where n is any integer.
sinθ = 0 → θ = nπ
cosθ = 0 → θ = (2n + 1)π/2
tanθ = 0 → θ = nπ
sinθ = sinα → θ = nπ + (−1)nα
cosθ = cosα → θ = 2nπ ± α
tanθ = tanα → θ = nπ + α
For example, to solve 2cosθ = 1, write cosθ = 1/2 = cos60° = cos(π/3). Hence θ = 2nπ ± π/3.
Do not stop at the principal value. Unless the question restricts the range (say 0 ≤ θ < 2π), you must give the general solution with n.
Range, Maximum and Minimum Values
NDA frequently asks for the greatest or least value of a trig expression. Two facts settle most of them.
−1 ≤ sinθ ≤ 1 and −1 ≤ cosθ ≤ 1
For a·sinθ + b·cosθ: maximum = √(a² + b²), minimum = −√(a² + b²)
So 3sinθ + 4cosθ ranges between −5 and +5, because √(3² + 4²) = √25 = 5. Also note that secθ and cosecθ never lie between −1 and 1; their values are always ≥ 1 or ≤ −1. This is why an equation like secθ = 1/2 has no solution at all — a fact examiners sometimes hide inside a multiple-choice trap.
Expressions of the form a + b·sinθ or a + b·cosθ are handled the same way: plug in the extreme values ±1 for the sine or cosine. For instance, 5 + 3cosθ has a maximum of 8 (when cosθ = 1) and a minimum of 2 (when cosθ = −1). No calculus required.
The minimum value of sin2θ or cos2θ is 0, not −1, because squaring removes the sign. Watch for the square.
Previous-Year Question and Quick Recap
Q. If tanθ + cotθ = 2, then the value of tan2θ + cot2θ is:
Answer: Square both sides: (tanθ + cotθ)2 = 4, so tan2θ + cot2θ + 2(tanθ·cotθ) = 4. Since tanθ·cotθ = 1, we get tan2θ + cot2θ + 2 = 4, hence tan2θ + cot2θ = 2.
- Six ratios: sin = P/H, cos = B/H, tan = P/B, and their reciprocals.
- Three identities: sin2+cos2=1, 1+tan2=sec2, 1+cot2=cosec2.
- Signs by ASTC; allied angles change the function only at 90°/270°.
- General solutions use n; a·sinθ+b·cosθ ranges within ±√(a²+b²).
- Memorise the 0°–90° table cold — it powers everything.
Frequently asked questions
How many marks does trigonometry carry in the NDA Maths paper?
Trigonometric ratios, identities, equations, inverse functions and height-and-distance together fetch roughly 25 to 30 marks, making trigonometry one of the largest and most scoring units in NDA Maths.
What is the fastest way to remember the standard-angle table?
Write 0, 1, 2, 3, 4 under 0 to 90 degrees, divide each by 4 and take the square root for the sine values. Reverse the order to get cosine, then derive tan as sin divided by cos.
What is the ASTC rule in trigonometry?
ASTC (All, Sin, Tan, Cos) tells you which ratios are positive in each quadrant: All in Quadrant I, only Sin in II, only Tan in III, and only Cos in IV. It is remembered as 'All Silver Tea Cups'.
When does a trigonometric ratio become undefined?
A ratio is undefined when its denominator is zero. For example tan and sec are undefined at 90 degrees because cos90 = 0, while cot and cosec are undefined at 0 degrees because sin0 = 0.
How do I write the general solution of a trigonometric equation?
Reduce the equation to sin, cos or tan of a known angle, then use the standard forms: sin gives n-pi plus (minus one) to the power n times alpha, cos gives 2n-pi plus or minus alpha, and tan gives n-pi plus alpha, where n is any integer.
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