Trigonometry

In a right-angled triangle with respect to angle θ:

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent = sin θ / cos θ
• Reciprocals: cosec θ = 1/sin, sec θ = 1/cos, cot θ = 1/tan.

Standard angles (memorise this row by row):

Pythagorean identities:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Complementary angles (always cross sin↔cos, tan↔cot, sec↔cosec):

• sin(90°−θ) = cos θ, cos(90°−θ) = sin θ
• tan(90°−θ) = cot θ, etc.

Sign rule (ASTC — All-Silver-Tea-Cups), as θ rotates from 0° to 360°:

• Quadrant I (0°–90°): All positive.
• Quadrant II (90°–180°): sin and cosec positive, others negative.
• Quadrant III (180°–270°): tan and cot positive, others negative.
• Quadrant IV (270°–360°): cos and sec positive, others negative.

Angle of elevation/depression:

• Elevation = angle from horizontal up to an object above.
• Depression = angle from horizontal down to an object below.
• Both are equal in magnitude when the observer and object swap (alternate-interior angle property).

Example 1 — Standard value substitution:
Find: 4 sin²30° + cos²60° − tan²45°
= 4 × (1/2)² + (1/2)² − 1² = 4·¼ + ¼ − 1 = 1 + ¼ − 1 = ¼.

Example 2 — Height of a tower:
A man 1.8 m tall stands 18 m from a tower. The angle of elevation of the tower's top from his eye is 45°. Height of tower?
Method: From eye level, height of tower above eye = 18 × tan 45° = 18 m. Total height = 18 + 1.8 = 19.8 m.

Example 3 — Identity simplification:
Simplify: (1 − sin²θ) × sec²θ.
= cos²θ × (1/cos²θ) = 1.

Example 4 — Two-angle problem:
A 30 m tower casts a 30√3 m shadow. Angle of elevation of the sun?
Method: tan θ = 30 / (30√3) = 1/√3 ⟹ θ = 30°.

Example 5 — Boat moving:
The angle of elevation of a cliff from a boat is 30°. The boat moves 100 m closer and the angle becomes 60°. Height of cliff?
Method: Let cliff height = h, original distance = d.
tan 60° = h/(d−100) ⟹ √3 = h/(d−100), so d−100 = h/√3.
tan 30° = h/d ⟹ 1/√3 = h/d, so d = h√3.
Subtract: 100 = d − (d−100) = h√3 − h/√3 = h(√3 − 1/√3) = h × 2/√3.
h = 100√3/2 = 50√3 ≈ 86.6 m.

Memory trick for sin row of standard table:
Write √0/2, √1/2, √2/2, √3/2, √4/2 → 0, 1/2, 1/√2, √3/2, 1. Cosine is the reverse order.