NDA Trigonometry: Important Identities, Formulas and Questions

NDA trigonometry covers ratios and identities, compound and multiple angles, inverse trigonometric functions, properties of triangles, and heights and distances. Learn the Pythagorean identities, the standard angle values, the sum and double-angle formulas, the inverse trig ranges and the sine and cosine rules. Then practise timed questions from Previous Year Papers.

You know sin²θ + cos²θ = 1. Then an NDA question asks for tan⁻¹(1/2) + tan⁻¹(1/3), and the clock keeps running while you try to remember the right formula.

This page puts every formula you need in one place, grouped by topic, with six solved questions that show the fastest method. Keep it open while you revise.

Trigonometry in the NDA Maths Paper

Trigonometry is one of the biggest units in Paper I, and it pays off quickly once the formulas are fixed in your head.

One question-bank publisher puts NDA trigonometry at 15 to 20 questions per paper. That is an estimate, not an official figure. Confirm the dates in the notification on upsc.gov.in.

Basic Identities and Standard Angle Values

Most NDA trigonometry questions come down to three Pythagorean identities and one table of values.

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ
• Radian measure: 180° = π radians

For angles beyond 90°, use the signs by quadrant: all ratios are positive in the first, only sin in the second, only tan in the third, and only cos in the fourth.

Compound, Multiple and Sub-Multiple Angle Formulas

These formulas turn awkward angles like 75° or 15° into standard ones. They also simplify most expression questions in the paper.

• Sum and difference: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
• Double angle formula: sin2A = 2 sinA cosA; cos2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1; tan2A = 2tanA / (1 − tan²A)
• Double angle in terms of tan: cos2A = (1 − tan²A) / (1 + tan²A)
• Triple angle: sin3A = 3sinA − 4sin³A; cos3A = 4cos³A − 3cosA
• Half angle: sin²(A/2) = (1 − cosA) / 2; cos²(A/2) = (1 + cosA) / 2

A guess on a formula you half remember costs you under negative marking, so write this list out from memory twice a week until it sticks.

Inverse Trigonometry Formulas

Inverse functions sit inside the NDA maths syllabus trigonometry unit, and most errors here come from forgetting the principal value range.

• Principal ranges: sin⁻¹x in [−π/2, π/2], cos⁻¹x in [0, π], tan⁻¹x in (−π/2, π/2)
• Complementary pairs: sin⁻¹x + cos⁻¹x = π/2 and tan⁻¹x + cot⁻¹x = π/2
• Reciprocal form: cot⁻¹x = tan⁻¹(1/x) for x > 0
• Sum of tangents: tan⁻¹x + tan⁻¹y = tan⁻¹((x + y) / (1 − xy)) when xy < 1

Before you mark an answer, check that it falls inside the range. An answer outside it is wrong even if the algebra looks right.

Properties of Triangles and Heights and Distances

This is the applied side of NDA trigonometry. A triangle-specific set of 2 to 4 questions per paper is a common estimate.

• Sine rule: a/sinA = b/sinB = c/sinC = 2R, where R is the circumradius
• Cosine rule: cosC = (a² + b² − c²) / 2ab
• Area: ½ ab sinC
• Elevation and depression: tan(angle) = height / horizontal distance

For heights and distances NDA questions, draw the figure first and mark the right angle. The angle of depression from the top equals the angle of elevation from the ground, because the two lines are parallel.

Solved NDA-Style Questions

Each question below uses one formula from above. Try it yourself first, then check the method.

Question 1: Using sinθ + cosθ

If sinθ + cosθ = √2, find sinθ cosθ.

Square both sides: sin²θ + cos²θ + 2 sinθ cosθ = 2. The first two terms add to 1, so 2 sinθ cosθ = 1 and sinθ cosθ = 1/2.

Question 2: Value of sin 75°

Write 75° as 45° + 30°. Then sin75° = sin45° cos30° + cos45° sin30° = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1) / 2√2. Rationalise to get (√6 + √2) / 4.

Question 3: Sum of inverse tangents

Find tan⁻¹(1/2) + tan⁻¹(1/3). Here xy = 1/6, which is less than 1, so the sum formula applies: tan⁻¹((1/2 + 1/3) / (1 − 1/6)) = tan⁻¹((5/6) / (5/6)) = tan⁻¹(1) = π/4.

Question 4: cos 2θ from tan θ

If tanθ = 3/4, find cos2θ. Use the tan form: cos2θ = (1 − 9/16) / (1 + 9/16) = (7/16) / (25/16) = 7/25. There is no need to find sinθ or cosθ separately.

Question 5: Height of a tower

From a point 30 m from the foot of a tower, the angle of elevation of the top is 60°. Then tan60° = h / 30, so h = 30√3, which gives 30√3 m (about 52 m).

Question 6: Angle from three sides

A triangle has sides 3, 5 and 7. Find its largest angle. The largest angle is opposite the longest side, so use the cosine rule with c = 7: cosC = (9 + 25 − 49) / 30 = −1/2. So C = 120°. A negative cosine tells you straight away that the angle is obtuse.

How to Prepare Trigonometry for NDA

• Build the base from NCERT: work through Class 11 trigonometry and Class 12 inverse functions before you move to exam-level questions.
• Practise by chapter: one exam-prep publisher suggests 30 to 40 question-bank problems per chapter. Keep a one-page sheet of formulas, and add the rest of the syllabus from our NDA maths formulas by chapter.
• Solve real papers: time yourself on trigonometry questions from NDA Previous Year Papers. Aim for under a minute per question.
• Use the options: put a standard angle such as θ = 45° into the expression and check which option matches.

FAQs

How many trigonometry questions come in NDA?
The number changes from paper to paper, and UPSC does not publish a fixed figure. The estimate above is a fair guide.

Is NCERT enough?
NCERT gives you the concepts. You still need timed practice with NDA-level questions.

Degrees or radians?
Both appear. Inverse trigonometry answers are usually given in radians.

Should I guess?
A blind guess barely pays off under negative marking. Guess only once you have ruled out at least one option.

Start preparing today: write the formula sheet from memory, solve the six questions again without looking, then take one timed paper. For free practice sets and exam updates, join the Defence Study community channels on WhatsApp and Telegram.