Basic Ratios (SOHCAHTOA)

sin θ = P/H; cos θ = B/H; tan θ = P/B; cosec θ = H/P; sec θ = H/B; cot θ = B/P

First step in any right-triangle problem — identify Perpendicular (P), Base (B) and Hypotenuse (H).

Standard Values Table

sin: 0°=0, 30°=½, 45°=1/√2, 60°=√3/2, 90°=1 | cos: reverse of sin row | tan: 0°=0, 30°=1/√3, 45°=1, 60°=√3, 90°=∞

Whenever a specific angle is given — substitute directly to get an exact numerical value.

Pythagorean Identities

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

Simplifying expressions and proving identities — convert any ratio to sin/cos then apply these.

Complementary Angle Pairs

sin θ = cos(90°−θ) | tan θ = cot(90°−θ) | sec θ = cosec(90°−θ)

When two angles in an expression add up to 90° — replace one ratio to create cancellable pairs.

Heights and Distances

tan(angle of elevation) = height / horizontal distance | angle of depression = angle of elevation (alternate interior angles)

Word problems with towers, poles, or observers looking up/down — draw a right triangle and apply tan.