Question
If tanθ = 3/4 and θ is acute, find sinθ and cosθ,
and verify that sin²θ + cos²θ = 1.Solution
Let initial distance from tower foot = x m, tower height = h m. From first position: tan45° = h/x ⇒ 1 = h/x ⇒ h = x From second position (distance x − 10): tan60° = h/(x − 10) ⇒ √3 = h/(x − 10) Substitute h = x: √3 = x/(x − 10) √3(x − 10) = x √3x − 10√3 = x (√3 − 1)x = 10√3 x = 10√3 / (√3 − 1) Rationalise: x = 10√3(√3 + 1) / [(√3 − 1)(√3 + 1)] = 10√3(√3 + 1) / (3 − 1) = 5√3(√3 + 1) = 5(3 + √3) Height h = x = 5(3 + √3) m.
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