Applied trigonometry practice

The problem bank

Every template takes your own measurements and reports the answer together with the steps that produced it. The first 10 are live for everyone; the remaining 25, marked Premium, are listed by name and open with Premium.

The one setup behind most of them

Elevation, depression, shadow and ladder problems are the same right triangle wearing different clothes. You stand a known horizontal distance from something and measure the angle to it, or you know the height and measure the angle down. Either way the two sides in play are the one across from the angle and the one beside it, so the tangent is the ratio you want.

Multiply through and a height is the distance times the tangent of the elevation; divide instead and a distance is the height over the tangent of the depression. The ladder problem swaps in sine and cosine because the ladder itself is the hypotenuse.

When the triangle has no right angle

Once the right angle goes, the two laws take over. Two angles and a side is AAS, which the law of sines closes in one step. Two sides and the angle between them is SAS, which the law of cosines closes. Three sides is SSS, which the same law rearranges into an arccosine for each angle, and which Heron turns into an area with no angle at all.

The one arrangement that needs care is SSA: an angle, the side opposite it, and one more side. The arcsine returns an acute angle, but its supplement shares that sine, so two different triangles can satisfy the same three measurements. The SSA template reports both whenever both exist, says so when only one survives the 180 degree sum, and says no triangle fits when the sine it needs would exceed 1. It never picks one and stays quiet.

Repeating motion

A wheel turning at a steady rate, a tide, and a plain transformed sine wave are the same model: an amplitude swinging either side of a midline, once every period. Boarding a wheel at its lowest point makes the height the centre height minus the radius times the cosine of the fraction of a turn completed, which is why the answer starts at the bottom rather than at the middle. Going the other way, from a height back to a time, is an inverse cosine, and the wheel passes every height twice in a turn: once climbing, once coming down.

Bearings and surveying

Bearings rotate the frame. A bearing is measured clockwise from north rather than counter-clockwise from east, so the northward component uses the cosine and the eastward component uses the sine. Getting that pair the wrong way round is the classic bearing mistake, and it always shows up as a route that is correct in length but reflected.

Once the components are right, navigation is addition. A two-leg course, a boat set off its heading by a current, an aircraft crabbing into a crosswind and two ships opening a gap from one port are all the same operation: resolve each piece into north and east, add, then recombine. Only the wind correction runs it backwards, solving for the heading that makes the sideways components cancel, which is why it can report that no heading works at all.

Surveying problems break the right-angle assumption altogether. Sighting one point from two ends of a measured baseline gives two angles and the side between them, which is an ASA triangle for the law of sines. If you would rather work that triangle yourself, the triangle solver takes the same three values and shows the full method.

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