Trigonometry (Identities, Inverse, Heights)
Syllabus Unit: Mathematics
Why it matters
Chapter overview
Trigonometry in the NDA syllabus builds from the six basic ratios and their core identities, through compound and multiple angle formulas that let you rewrite any trig expression, to inverse trigonometric functions with their restricted principal values, and finally height-and-distance word problems that are trigonometry's most direct real-world application and a guaranteed few marks every year.
Mathematics & General Ability Mathematics 40% weightage
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sin 30°
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sin 45°
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sin 60°
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sin 90°
1Trigonometric Ratios & Identities
Six ratios of a right triangle's sides generate an entire web of identities that let you rewrite any trig expression.
Learning Objectives
- Define sin, cos, tan, cosec, sec, cot for a right triangle and recall their values at 0°, 30°, 45°, 60°, 90°.
- State and apply the Pythagorean identities: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=cosec²θ.
- Determine the sign of each ratio in the four quadrants using the ASTC rule.
- Convert between degrees and radians.
Key Terms
Pythagorean Identitysin²θ + cos²θ = 1, derived directly from the Pythagorean theorem on the unit circle.
ASTC Rule'All Sin Tan Cos' — mnemonic for which ratios are positive in quadrants I, II, III, IV respectively.
RadianThe angle subtended by an arc equal in length to the radius; π radians = 180°.
Reciprocal Ratioscosecθ=1/sinθ, secθ=1/cosθ, cotθ=1/tanθ.
The six trigonometric ratios relate a right triangle's angle to the ratios of its sides: sin, cos, tan and their reciprocals cosec, sec, cot.
Standard angle values
- sin: 0, 1/2, 1/√2, √3/2, 1 at 0°,30°,45°,60°,90°
- cos: 1, √3/2, 1/√2, 1/2, 0 at the same angles
- tan: 0, 1/√3, 1, √3, undefined at the same angles
Core identities
sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ
Exam TipThe second and third identities follow just by dividing the first through by cos²θ or sin²θ — no need to memorise all three separately.
Quadrant signs (ASTC)
- Quadrant I: all ratios positive.
- Quadrant II: only sin (and cosec) positive.
- Quadrant III: only tan (and cot) positive.
- Quadrant IV: only cos (and sec) positive.
Common MistakeSign errors for angles like 120°, 210°, 300° are one of the most common NDA trig mistakes — always reduce to the nearest acute angle first.
Degrees vs radians
θ (radians) = θ (degrees) × π/180