Trigonometry (Identities, Heights & Distances)

Trigonometry (Identities, Heights & Distances)

Concept

Trigonometry in SSC Tier 2 has two components: algebraic manipulation using trigonometric identities, and practical application in heights-and-distances word problems. The algebraic side tests your ability to simplify expressions using standard identities, while the geometry side tests your ability to model real situations as right triangles.

Standard Values to Memorise:

Angle	0°	30°	45°	60°	90°
sin	0	1/2	1/√2	√3/2	1
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	∞

Key Identities:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ (for θ ≠ 90°)
• 1 + cot²θ = cosec²θ (for θ ≠ 0°)
• sin(A+B), sin(A−B), cos(A+B), cos(A−B) expansion formulas
• Max/min values: For a sin θ + b cos θ, range is [−√(a²+b²), √(a²+b²)]

Heights and Distances:

• Angle of elevation: when looking UP from horizontal.
• Angle of depression: when looking DOWN from a height — equals angle of elevation from the lower point to the higher point.
• Always draw the right triangle — mark known sides, mark unknown (what you’re solving for).

Key Points

• tan θ = sin θ / cos θ. cot θ = cos θ / sin θ = 1/tan θ.
• sec θ = 1/cos θ. cosec θ = 1/sin θ.
• Complementary angles: sin θ = cos(90°−θ), tan θ = cot(90°−θ).
• For heights and distances, if the angle and one side are known, use the appropriate trig ratio to find the other side.
• When two angles of elevation are given from two different points to the same object, use tan values to set up two equations.

Worked Example

Q: The angle of elevation of the top of a tower from a point 30m away from its base is 30°. Find the height of the tower. Approach: tan 30° = Height / 30 → 1/√3 = h/30 → h = 30/√3 = 10√3 m. Answer: 10√3 m

SSC Pattern / Tips

• In heights-and-distances, always check whether the point is on level ground with the base or above/below it.
• When an object is viewed from two different points, use both angles to create two equations.
• For identity simplification, try to convert everything to sin and cos first — it’s often the most general approach.
• Maximum/minimum value of expressions like sin θ + cos θ = √2 × sin(θ+45°), so max = √2.