Concept 1
Name every side relative to the chosen angle
In a right triangle, the side opposite the 90° angle is the hypotenuse and is always the longest side. For an acute angle θ, the other sides are the opposite (perpendicular) and adjacent (base) sides.
The labels depend on θ: when the reference angle changes, opposite and adjacent interchange; the hypotenuse does not.
Pythagoras connects the sides: opposite²+adjacent²=hypotenuse².
Concept 2
The six ratios compare two sides
| Ratio | Definition | Memory link |
|---|---|---|
| sin θ | opposite / hypotenuse | O/H |
| cos θ | adjacent / hypotenuse | A/H |
| tan θ | opposite / adjacent | O/A |
| cosec θ | hypotenuse / opposite | 1/sin θ |
| sec θ | hypotenuse / adjacent | 1/cos θ |
| cot θ | adjacent / opposite | 1/tan θ |
SOH–CAH–TOA remembers the three primary ratios. Use a reciprocal only after identifying its paired primary ratio.
Concept 3
Know the standard-angle table exactly
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
| cosec θ | undefined | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | undefined |
| cot θ | undefined | √3 | 1 | 1/√3 | 0 |
For sin, read √0/2, √1/2, √2/2, √3/2, √4/2. Cos uses the same sequence in reverse. Then tan=sin/cos.
Concept 4
Reciprocal and quotient relations reduce memorisation
- sin θ·cosec θ=1, cos θ·sec θ=1, and tan θ·cot θ=1 wherever defined.
- tan θ=sin θ/cos θ.
- cot θ=cos θ/sin θ.
“Undefined” is not a very large number. It occurs when the ratio would require division by zero.
Concept 5
Three identities organise most simplifications
| Identity | Useful rearrangement |
|---|---|
| sin²θ+cos²θ=1 | 1−sin²θ=cos²θ |
| 1+tan²θ=sec²θ | sec²θ−tan²θ=1 |
| 1+cot²θ=cosec²θ | cosec²θ−cot²θ=1 |
When signs matter beyond acute-angle questions, a squared identity may determine only the magnitude. In the SSC right-triangle scope, θ is ordinarily acute unless the question says otherwise.
Concept 6
Co-functions swap at complementary angles
- sin(90°−θ)=cos θ and cos(90°−θ)=sin θ.
- tan(90°−θ)=cot θ and cot(90°−θ)=tan θ.
- sec(90°−θ)=cosec θ and cosec(90°−θ)=sec θ.
A complement changes the function to its co-function; it does not change the value.
Concept 7
Lines of sight create the angle
The angle of elevation is measured upward from the observer’s horizontal line to an object above it. The angle of depression is measured downward from the horizontal at the higher point.
Because the two horizontal lines are parallel, an angle of depression equals the corresponding angle of elevation. Draw the horizontal explicitly before using that equality.
Concept 8
Height-and-distance problems follow four moves
- Draw the object vertical and the ground horizontal, creating a right angle.
- Mark the observer, line of sight, angle and all known distances.
- Adjust for eye height when the observer is above ground.
- Choose the ratio containing the known and required sides—often tan θ=height/horizontal distance.
Level ground: if a tower of height h is viewed from horizontal distance d at elevation θ, then h=d tan θ. If eye height e is given, total height=e+d tan θ.
Worked methods
Translate first, calculate second
Find a ratio from sides
For a 3–4–5 triangle and θ opposite side 3: sin θ=3/5, cos θ=4/5 and tan θ=3/4.
Recover remaining ratios
If tan θ=3/4, choose opposite=3 and adjacent=4. Hypotenuse=5, so sin θ=3/5 and sec θ=5/4.
Use an identity
If sin θ=5/13 for acute θ, cos θ=√(1−25/169)=12/13.
Single observation
A pole is 20 m away at elevation 45°. Height=20 tan45°=20 m.
Include eye height
From eye level 1.6 m, a building top is seen at 30° from 30√3 m. Rise=30√3×1/√3=30 m. Total=31.6 m.
Two positions on one line
A tower has elevations 60° and 30° from points 20 m apart on the same side. If nearer distance=x, x√3=(x+20)/√3, so 3x=x+20, x=10 and height=10√3 m.
Exam traps
Protect the diagram from hidden assumptions
Wrong reference angle
Opposite and adjacent must be named relative to the marked θ.
Hypotenuse misidentified
It lies opposite 90°, not necessarily in a particular drawing position.
Depression measured from vertical
It is measured from a horizontal line at the observer.
Eye height omitted
The triangle gives height above the eye, not always total object height.
Distance along sight line
Ground distance is adjacent; line-of-sight distance is the hypotenuse.
Decimal too early
Keep √3 and fractional ratios exact until the final step.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute formula map
- SOH–CAH–TOA: sin=O/H, cos=A/H, tan=O/A.
- cosec=1/sin, sec=1/cos, cot=1/tan; tan=sin/cos.
- sin values from 0° to 90°: 0, 1/2, 1/√2, √3/2, 1; cos reverses them.
- tan 0°=0, tan30°=1/√3, tan45°=1, tan60°=√3, tan90° undefined.
- sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ.
- Complement pairs: sin↔cos, tan↔cot, sec↔cosec.
- Elevation and depression are measured from horizontal lines.
- On level ground, height above eye=d tan θ; add eye height if required.
Source ledger
Official foundations
- SSC CGL official syllabus: Trigonometry and heights-and-distances scope
- NCERT official textbook portal: Introduction to Trigonometry and its applications
- NCERT official exemplar directory
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 29 September 2026.
Tables, charts and statistics.