Revise: Trigonometry

sinθ=y/r, cosθ=x/r, tanθ=y/x, for any point (x,y) at distance r.

P(−3,4) → r=5, sinθ=0.8.

ASTC: I all+, II sin+, III tan+, IV cos+.

cos−, sin+ → Quadrant II.

sin(−θ)=−sinθ, cos(−θ)=cosθ, tan(−θ)=−tanθ.

sin40°≈0.643 → sin(−40°)≈−0.643.

Reduce any angle to an acute reference angle, then apply ASTC sign.

sin150° = +sin30° = 0.5.

tan undefined at 90°, 270° (x=0 there).

sin270°=−1.

sin²θ + cos²θ = 1, always.

sinθ=0.6 → cosθ=0.8 (QI).

tanθ = sinθ/cosθ.

sin=0.6, cos=0.8 → tan=0.75.

a/sinA = b/sinB = c/sinC.

d=5,D=30°,E=90° → e=10.

c² = a²+b²−2ab·cosC.

a=7,b=9,C=60° → c≈8.19.

Bearings: clockwise from north; reverse bearing = ±180°.

40km@065°, 30km@155° → AC=50km.

Area = ½ab·sinC.

a=8,b=5,C=30° → Area=10.

Matches right-triangle trig exactly in QI; x,y can be negative elsewhere.

P(5,−12) → r=13.

tan's sign is sin's sign × cos's sign combined.

tan+, cos− → Quadrant III.

Reflecting across the x-axis flips y (sin, tan) but not x (cos).

cos25°≈0.906 → cos(−25°)≈0.906.

QII: α=180−θ. QIII: α=θ−180. QIV: α=360−θ.

cos210° = −cos30° ≈ −0.866.

Values follow directly from the general definition at axis points.

tan180°=0; tan90° undefined.

Comes from the Pythagorean theorem on the unit circle (r=1).

cosθ=−0.28 (QIII) → sinθ=−0.96.

Follows from (y/r)/(x/r) = y/x.

sin=0.8, cos=0.6 → tan≈1.33.

Derived by writing one height two different ways.

G=H=45°, g=8 → h=8.

Reduces to Pythagoras exactly when C=90° (cosC=0).

p=8,q=6,R=90° → r=10.

Convert the bearing description into a triangle's interior angle first.

Bearing 040° → reverse bearing 220°.

Derived from h=a·sinC substituted into ½×base×height.

a=10,b=12,C=90° → Area=60.