Pythagoras and Trigonometry

One decision, then the tool

Most people freeze in front of a triangle question because they think the hard part is the calculation. It is not. The hard part is choosing the right tool, and that is one quick decision. ⭐ Ask one thing first: is an angle involved? If the question is only about the three sides, with no angle anywhere in it, the tool is Pythagoras. The moment an angle is in the question, whether you are given it or asked to find it, the tool is trigonometry. ⭐⭐ So the whole skill is this: LABEL WHAT YOU HAVE, and LABEL WHAT YOU WANT. Sides only means Pythagoras. Anything with an angle means trigonometry, and the sides in the question then tell you which ratio. ⚠️ Carry that one question through everything below. Not what is the answer, but which tool, and the answer follows.

When to reach for each tool

The two tools of this topic do different jobs. Keeping them apart is the whole of the decision.

Order a right-triangle solve

Put the steps of working any right-angled-triangle question into the order that gets you to the answer.

• Label what you already have and what you want to find
• Decide whether an angle is involved at all
• Choose Pythagoras if it is sides only, or trigonometry if there is an angle
• Put the numbers you have into the rule for that tool
• Solve for the missing length or angle

Which tool does it need

A right-angled triangle has two sides given, 6 cm and 8 cm, and you are asked for the third side. No angle is mentioned. Which tool does this need?

• Pythagoras theorem, because the question is only about sides and no angle is involved
• The sine ratio, because triangles need trigonometry
• The tangent ratio, because there are two sides
• The cosine rule, because there are three sides in play

Words for right-angled triangles

Five terms, each defined so you can use it in a sentence rather than only recognise it.

Find the hypotenuse

A right-angled triangle has its two shorter sides measuring 3 cm and 4 cm. Using Pythagoras theorem, the square of the hypotenuse equals 3 squared plus 4 squared. What is the length of the hypotenuse, in centimetres?

True of these tools

Select the TWO statements that are true.

• Pythagoras theorem is used when no angle is involved, only the three sides
• Which trig ratio you use depends on which two sides are in the question
• The hypotenuse is the shortest side of a right-angled triangle
• Every triangle question must be solved with trigonometry

Choosing the ratio with SOH CAH TOA

Once you know an angle is involved and it is trigonometry, one short routine picks the ratio. ⭐ Label the three sides from the angle you are using. The side opposite that angle, the side next to it, and the hypotenuse. ⭐ See which two sides the question involves. One you have, one you want. It is always two of the three. ⭐ Read SOH CAH TOA. Sine uses the Opposite and the Hypotenuse. Cosine uses the Adjacent and the Hypotenuse. Tangent uses the Opposite and the Adjacent. ⭐ Pick the ratio whose two letters are the two sides you have and want. That is the ratio, with no guessing. ⚠️ A missing SIDE and a missing ANGLE use the same three ratios; finding an angle just means working the ratio backwards. The choice of ratio is made the same way either time. And remember the gate before all of this: if there is no angle at all, you are not here, you are back at Pythagoras.

Match the case to the tool

Find the opposite side

A right-angled triangle has a hypotenuse of 10 cm and an angle of 30 degrees. The side opposite that angle equals the hypotenuse multiplied by the sine of the angle. Given that the sine of 30 degrees is 0.5, how long is the opposite side, in centimetres?

Build the decision rule

Assemble the one rule that decides every right-angled-triangle question.

A Higher tool: the area of a triangle

At Higher you meet triangles that are not right-angled, and a neat formula for their area. The area of a triangle is one half of one length, times another length, times the sine of the angle between them. ⭐ Take a triangle with two lengths of 8 cm and 5 cm, and an angle of 30 degrees between them. The sine of 30 degrees is 0.5. Area equals one half times 8 times 5 times 0.5. One half of 8 is 4, times 5 is 20, times 0.5 is 10. ⭐⭐ So the area is 10 square centimetres, and notice you never needed a right angle at all. The angle between the two lengths did the work. The same family of Higher tools includes the sine rule and the cosine rule, which find missing lengths and angles in triangles that are not right-angled. They are the same decision as before, made on a triangle without a right angle in it.

Complete the triangle facts

In a right-angled triangle, the longest side, opposite the right angle, is the _____. Pythagoras theorem says the square of that side equals the sum of the squares of the other two _____. When an angle is involved you use trigonometry, whose three ratios are sine, _____ and tangent. The side across the triangle from the angle you are using is the _____ side, and the side beside that angle, which is not the hypotenuse, is the _____ side.

The triangle tool run

Five quick decisions on choosing and using the right tool. Three lives.

Work a two-step problem

A ladder leans against a wall. You are told the foot of the ladder is 5 cm from the wall on a scale drawing and the ladder is 13 cm long, and later you are asked for the angle the ladder makes with the ground. Work through the tools.

• First you want the height the ladder reaches up the wall. You have two sides, the 5 and the 13, and no angle yet. Which tool?
• Now you want the angle the ladder makes with the ground. You have sides and you want an angle. Which family of tools?
• You decide to use the side opposite the ground angle and the hypotenuse. Which ratio does SOH CAH TOA give?

Explain how to choose the tool

A friend says they can never tell whether a triangle question needs Pythagoras or trigonometry, and they guess. Explain to them how to decide every time.

• Explain the one question to ask first: is an angle involved
• Explain when the answer means Pythagoras, and what that tool links
• Explain when the answer means trigonometry, and how SOH CAH TOA then picks the ratio
• Use the terms hypotenuse, opposite and adjacent correctly
• Finish with the idea that labelling what you have and what you want makes the tool choose itself