Trigonometry Questions for NDA

Trigonometric Ratios & Standard Angles

The six trigonometric ratios (sin, cos, tan, cot, sec, cosec) are the foundation. In NDA, you must instantly recall values for 0, 30, 45, 60, 90 degrees — these appear in almost every paper. Remember: sin and cos are 'opposite' functions (sin 0 = 0, cos 0 = 1; sin 90 = 1, cos 90 = 0). Tan is the 'slope' ratio (opposite/adjacent). NDA often tests whether you can convert between degrees and radians, and apply these ratios in right triangles.

• sin(0°)=0, sin(30°)=1/2, sin(45°)=1/√2, sin(60°)=√3/2, sin(90°)=1
• cos values are reverse: cos(0°)=1, cos(30°)=√3/2, cos(45°)=1/√2, cos(60°)=1/2, cos(90°)=0
• tan(0°)=0, tan(30°)=1/√3, tan(45°)=1, tan(60°)=√3, tan(90°)=undefined
• Radian conversion: 180° = π radians; multiply degrees by π/180 to convert
• ASTC rule: All positive in Quadrant I; Sin positive in Q2; Tan positive in Q3; Cos positive in Q4

Trigonometric Identities & Simplification

NDA loves identity-based questions because they test conceptual understanding, not just memorization. The key identities are: sum/difference formulas, double angle formulas, and product-to-sum conversions. These appear in almost every paper, often disguised in 'prove that' or 'simplify' questions. Learn to recognize patterns — e.g., sin(A+B) vs sin(A-B) — and apply them mechanically. The double angle formulas are especially high-yield because they compress two angles into one.

• Sum formulas: sin(A±B) = sinA·cosB ± cosA·sinB; cos(A±B) = cosA·cosB ∓ sinA·sinB
• Double angle: sin(2θ) = 2sinθ·cosθ; cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ; tan(2θ) = 2tanθ/(1-tan²θ)
• Product-to-sum: sinA·sinB = [cos(A-B) - cos(A+B)]/2; cosA·cosB = [cos(A-B) + cos(A+B)]/2
• Sum-to-product: sinA + sinB = 2sin[(A+B)/2]·cos[(A-B)/2]
• Complementary angles: sin(90°-θ) = cosθ; cos(90°-θ) = sinθ; tan(90°-θ) = cotθ

Inverse Trigonometric Functions

NDA tests inverse trig (arcsin, arccos, arctan) in two ways: (1) finding the angle given a ratio, and (2) simplifying expressions like sin(arctan(x)). The key is understanding the range: arcsin and arctan return angles in [-π/2, π/2]; arccos returns [0, π]. Many aspirants forget these ranges and lose marks. Also, sin(arctan(x)) = x/√(1+x²) — this type of conversion is a favorite NDA trick.

• arcsin(x) returns angle in [-π/2, π/2]; arccos(x) returns angle in [0, π]; arctan(x) returns angle in (-π/2, π/2)
• sin(arctan(x)) = x/√(1+x²); cos(arctan(x)) = 1/√(1+x²); tan(arcsin(x)) = x/√(1-x²)
• arcsin(x) + arccos(x) = π/2 for all x in [-1, 1]
• arctan(x) + arctan(y) = arctan[(x+y)/(1-xy)] when xy < 1
• Domain restrictions: arcsin and arccos only defined for x in [-1, 1]; arctan defined for all real x

Heights & Distances (Applications)

This is the 'story problem' section of trigonometry in NDA. You're given a scenario (observer on a hill, angle of elevation/depression, distance), and must find height or distance. The method is always the same: draw a diagram, identify the right triangle, pick the correct ratio (sin, cos, or tan), and solve. NDA examiners test whether you can translate words into geometry. Common traps: confusing angle of elevation with depression, or forgetting to add/subtract initial heights.

• Angle of elevation: angle above horizontal when looking UP at an object
• Angle of depression: angle below horizontal when looking DOWN at an object
• Key insight: angle of elevation from A to B = angle of depression from B to A (alternate angles)
• Always draw a diagram first; identify the right triangle and the known/unknown sides
• Use tan(θ) = opposite/adjacent when you have horizontal and vertical distances; use sin/cos when hypotenuse is involved

• Forgetting ASTC rule: Many aspirants compute sin(120°) as -√3/2 instead of +√3/2. Remember: sine is positive in Q1 and Q2. 120° is in Q2, so sin(120°) = sin(60°) = √3/2.
• Mixing up angle of elevation and depression: Elevation is UP (positive angle above horizontal), depression is DOWN (negative angle below horizontal). A common NDA trap is to use the wrong angle in the formula.
• Incorrect range for inverse functions: arcsin(0.5) = 30° OR 150°? No — arcsin only returns [-90°, 90°], so arcsin(0.5) = 30° only. The other solution is found separately using ASTC.
• Forgetting to simplify after identity application: You correctly apply sin(A+B) but then leave the answer unsimplified. NDA expects fully simplified final answers. Always check if you can combine like terms or use another identity.
• Careless radian-degree conversion: Converting 45° to radians as 45π instead of π/4. Always divide by 180 first: 45 × (π/180) = π/4.

• ASTC = All Students Take Calculus (or 'All Sin Tan Cos'): Quadrant I all positive, Q2 sine positive, Q3 tangent positive, Q4 cosine positive.
• SOH-CAH-TOA = Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Drill this until it's automatic.
• Magic angles: 0-30-45-60-90. Their sin/cos values follow a pattern: sin goes 0, 1/2, 1/√2, √3/2, 1 (increasing); cos is reversed (decreasing). Visualize a rising and falling curve.
• Complementary angles sum to 90°: sin(θ) = cos(90°-θ). This is why sin(30°) = cos(60°). Use this to check your answers.
• Double angle = 2 × single: sin(2θ) = 2sinθ·cosθ is literally 'two times' the single angle formula. Helps you remember it's not sin(θ)·sin(θ).

• NDA Maths papers (2019-2024) show a consistent pattern: 1-2 questions on standard angle values, 2-3 on identity simplification/proof, 1-2 on inverse functions, and 1-2 on heights/distances. Total ~8-10% of paper. Prioritize identities and standard angles.
• Identity questions are almost always 'Prove that' or 'Simplify'. They test whether you can chain multiple identities together. Practice converting between sin/cos/tan forms and recognizing when to apply sum vs. double angle formulas.
• Heights and distances questions are usually straightforward IF you draw the diagram correctly. Spend 30 seconds on the diagram; it prevents 90% of errors. NDA examiners reward clear working here.
• Inverse trig questions often hide in 'find the angle' or 'simplify sin(arctan(x))' format. Know the three key conversions (sin/cos/tan of arctan, arcsin, arccos) cold. These appear almost every year.
• Time management: Trig questions are medium-difficulty. Allocate 8-10 minutes for 2-3 trig questions. If you get stuck on an identity, move on and come back; don't waste 5 minutes on one proof.