Exact and calculator ratios

To solve a trigonometric equation, first isolate the ratio, then recognise the angle. 2 sin θ = 1 gives sin θ = 1/2, so θ = 30°. tan θ = √3 gives θ = 60°. If the value is not in our table, use the inverse key. For heights and distances, draw a right triangle: the angle of elevation is measured from the horizontal at the observer’s eye. Pick the ratio that holds the known side and the wanted side: opposite and adjacent → tan; opposite and hypotenuse → sin; adjacent and hypotenuse → cos. Example: standing 12 m from a tree, the angle of elevation of the top is 30°. Height = 12 × tan 30° = 12/√3 = 4√3 ≈ 6.93 m (ignoring the observer’s height).