Trigonometry: Identities and Heights and Distances
Ratios and identities
In a right triangle, sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse, tanθ=opposite/adjacent; cosec, sec, cot are reciprocals. The Pythagorean identity sin²θ+cos²θ=1 gives 1+tan²θ=sec²θ and 1+cot²θ=cosec²θ. Complementary angles: sin(90°−θ)=cosθ, tan(90°−θ)=cotθ, sec(90°−θ)=cosecθ.
Heights and distances
Draw a right triangle with the horizontal, mark the angle of elevation upward from the horizontal or the angle of depression downward from the horizontal. Then choose the ratio that connects the known and unknown sides. For a 30–60–90 triangle the sides are in ratio 1:√3:2; for 45–45–90 they are 1:1:√2.
Worked example
A ladder 10 m long leans against a vertical wall, making 60° with the ground. The height reached is the side opposite 60°, so h=10 sin60°=10×(√3/2)=5√3≈8.66 m. The distance of the foot from the wall is 10 cos60°=5 m.
Traps
Using the angle with the vertical instead of the horizontal; putting the known side in the wrong ratio; forgetting that sin²θ means (sinθ)²; and cancelling terms inside sin or cos as if they were factors. Check that the answer is sensible: the height cannot exceed the ladder length.