Trigonometry game: sine, cosine, tangent and right-angled triangles
A maths game for learning trigonometry without mixing up the sides. A right-angled triangle drawn to scale appears, with angle α marked sometimes at the bottom corner and sometimes at the top, and the side lengths you need. There are no answers to pick from: you build the sine, cosine or tangent by tapping the top and bottom sides of the fraction, tap each side to name it (opposite, adjacent, hypotenuse), complete the 30°, 45° and 60° table, and type a side by Pythagoras, a side or an angle with a calculator and the answers to word problems: a ladder against a wall, a tree’s shadow, a kite or a ramp. When you answer, each side is coloured with its name relative to α.
Mathematics · Upper secondary, Sixth form
The most common trigonometry mistake is learning the opposite side as “the one on the right” or “the vertical one”. Because the angle moves corner from round to round, you have to check each time which side faces α. When you answer, the hypotenuse turns yellow, the opposite side pink and the adjacent side blue.
The wrong answers are the ones you get by using the wrong ratio, multiplying when you should divide (and vice versa), adding the legs in Pythagoras or giving the complementary angle. Each mistake comes with the correct working and the ratio you needed.
In the word problems the triangle does not appear until you answer: turning “a 6 m ladder makes 65° with the ground” into hypotenuse, angle and opposite side is the hardest part in exams, and here you practise it again and again.
"A triangle has legs 8 and 15 and hypotenuse 17, with α at the top corner. You have to build cos α. Most people tap the horizontal side, 8, first and make 8/17. But with α at the top, the side facing α is the horizontal one (8, the opposite side) and the side touching it is the vertical one (15, the adjacent side). So cos α = adjacent ÷ hypotenuse = 15/17. When you answer, the 15 turns blue with the word “adjacent”."
Prefer to learn without screens? Try this analogue version:
Make a clinometer with a protractor, a straw, a thread and a small weight, or use a spirit-level app on a phone.
Stand a known distance from a building or tree and measure the angle up to the top.
Work out the height with the tangent: height = distance · tan(angle) + the height of your eyes.
Repeat from another distance and compare: if the two heights are close, you measured well.
If you like Right Angle, these activities work similar skills:
Another maths game: read the graph and adjust the function until it fits.
Solve first-degree equations with the balance method, the same rearranging you need to find a side.
Measure three tall things near you with a home-made clinometer and the tangent, then check one of them with a tape measure.
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