NDA Trigonometry: Identities, Angles, Heights and Distances

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Angles and measures in degrees and radians; trigonometric ratios; identities; sum and difference formulae; multiple and sub-multiple angles; inverse trigonometric functions; heights and distances; properties of triangles.

Angles and measures in degrees and radians; trigonometric ratios; identities; sum and difference formulae; multiple and sub-multiple angles; inverse trigonometric functions; heights and distances; properties of triangles.

Right-triangle geometry, algebraic manipulation, coordinate signs, basic similarity.

• Radian and degree measure
• Sine, cosine and tangent
• Quadrant signs
• Principal-value idea for inverse trigonometric functions
• Angle of elevation / depression

• sin^2 x + cos^2 x = 1
• 1 + tan^2 x = sec^2 x; 1 + cot^2 x = csc^2 x
• sin(A +/- B)=sinA cosB +/- cosA sinB
• cos(A +/- B)=cosA cosB -/+ sinA sinB
• tan(A +/- B)=(tanA +/- tanB)/(1 -/+ tanA tanB), denominator non-zero
• sin 2A=2sinA cosA; cos 2A=cos^2 A - sin^2 A
• Degree-radian: theta radians = theta degrees x pi/180
• Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: c^2=a^2+b^2-2ab cos C

• Tangent identities require denominators to be non-zero.
• Inverse-trig simplification must respect principal-value ranges.
• Height-distance sketches require a horizontal reference and correct elevation / depression angle.
• Triangle rules apply to the corresponding opposite sides and angles.

1. Convert 150 degrees to radians: 150 x pi/180 = 5pi/6.
2. If tan theta=3/4 in an acute angle, use a 3-4-5 triangle to get sin theta=3/5 and cos theta=4/5 without calculating theta.
3. A tower seen at elevation 45 degrees from distance 40 m on level ground has height 40 m if observer height is ignored. The diagram is part of the method, not decoration.

• Using degree values directly in a radian-based identity.
• Ignoring quadrant signs.
• Cancelling terms across addition in identities.
• Using inverse-trig identities outside principal ranges.
• Drawing height-distance geometry without marking horizontal / vertical lines.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Reduce the expression before substituting numbers. For triangle / height problems, draw and label first. For identities, convert to sine and cosine only when it simplifies structure. Under negative marking, avoid long identity transformations unless each step has a clear target.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.