Trig Trek

Trig Trek

Right-angled triangles have two tools, and picking between them takes one question: Does the problem involve an angle? If no, you have sides only, and it is Pythagoras. If yes, it is trigonometry, the sin, cos and tan you reach through SOHCAHTOA. That is the whole decision. Nearly every mark lost on this topic is lost by reaching for the wrong one of the two, so ask the question before you write anything.

Naming the sides

Two of these three names depend on which angle you are working from, and that is where the marks go:

Tap the hypotenuse

The small square marks the right angle. Tap the hypotenuse.

Add or subtract?

a² + b² = c², with c the hypotenuse. Everything depends on whether the side you want is the hypotenuse or not:

Find the hypotenuse

A right-angled triangle has legs of 6 and 8. How long is the hypotenuse?

Find the missing leg

A right-angled triangle has a hypotenuse of 13 and one leg of 5. How long is the other leg?

Spot the impossible answer

A student is finding the hypotenuse of a triangle with legs 9 and 12. They write down 10.5. Without redoing the calculation, how can you tell straight away that it is wrong?

• The hypotenuse is the longest side of the triangle, so it must be greater than 12. Any answer below 12 is impossible before you check a single digit of the arithmetic
• Side lengths always come out as whole numbers, so a decimal answer must be wrong
• The hypotenuse should be the average of the two legs, which is 10.5, so actually it is right
• The hypotenuse should equal 9 + 12 = 21

SOHCAHTOA

Three ratios link an angle to two of the sides: • S O H: sin θ = Opposite ÷ Hypotenuse • C A H: cos θ = Adjacent ÷ Hypotenuse • T O A: tan θ = Opposite ÷ Adjacent Choose by which two sides your question involves, not by which letter you remember best. If the question names the adjacent and the hypotenuse, it is cos, and there is nothing to decide. Worked example. Angle 60°, hypotenuse 12, find the adjacent. Adjacent and hypotenuse means cos, so cos 60° = adjacent ÷ 12. Rearranging, adjacent = 12 × cos 60° = 12 × 0.5 = 6.

Which tool for which pair?

Four things that cost marks

Your calculator is in the wrong mode. It must say DEG. In radians, sin 30 gives −0.988 instead of 0.5, and every answer afterwards is quietly wrong. You want an angle, so you need the inverse. sin, cos and tan turn an angle into a ratio. To go back the other way you need sin⁻¹, cos⁻¹ or tan⁻¹, the buttons usually reached via SHIFT. Exact values worth knowing by heart, because they let you check a calculator answer: • sin 30° = 0.5, and cos 60° = 0.5 • tan 45° = 1, since at 45° the two legs are equal • sin 90° = 1, and cos 90° = 0 Sanity-check the size. sin and cos of any angle in a right-angled triangle are always between 0 and 1, because you are dividing by the longest side. If a calculation hands you sin θ = 1.4, you have put a side in the wrong place.

Find the opposite side

In a right-angled triangle the angle θ is 30° and the hypotenuse is 20. How long is the side opposite θ?

Rearranging the ratio

You know the angle θ and the length of the adjacent side, and you want the opposite. Starting from tan θ = opposite ÷ adjacent, which rearrangement is right?

• opposite = adjacent × tan θ
• opposite = adjacent ÷ tan θ
• opposite = tan θ ÷ adjacent
• opposite = tan⁻¹(adjacent)

From two sides to an angle

A ladder reaches 4 m up a wall with its foot 3 m from the base. You want the angle it makes with the ground. You know the opposite and the adjacent, so you write tan θ = 4 ÷ 3 = 1.333. What is the next step, and roughly what answer should you expect?

• Apply tan⁻¹ to 1.333 to get θ, and expect something above 45°, since the opposite is longer than the adjacent
• Multiply 1.333 by the hypotenuse to get θ
• Use Pythagoras on the 3 and the 4 to get the angle
• Apply tan⁻¹ to 1.333, and expect an answer below 45°

Order the method

Put the steps for finding an unknown ANGLE from two known sides into the right order.

• Mark the angle you want as θ, then label the three sides from it
• Note which two sides you actually know, and pick the ratio built from those two
• Write the ratio as a division and work it out as a decimal
• Apply the inverse of that ratio to the decimal to turn it back into an angle
• Check the angle is plausible, and that the three angles could still total 180°

The trek rules

Ask first whether the problem involves an _____. If it does not, use Pythagoras: a² + b² = _____, where c is the hypotenuse, and remember to add when the missing side is the hypotenuse and subtract when it is a leg. If it does, label the sides from that angle and pick the ratio built from the two sides you have: sin is opposite over _____. To turn two sides back into an angle you need the _____ function, such as tan⁻¹.