Trig Trek

Trig Trek 🧭

Right-angled triangles have two trusty tools. **Pythagoras' theorem** finds a missing **side** when you know the other two. **Trigonometry** (SOHCAHTOA) handles problems with an **angle**. Learn to spot which tool a question needs, and no right-angled triangle can stop you.

The hypotenuse 📐

First, names. The **hypotenuse** is the **longest** side of a right-angled triangle — always the one **opposite the right angle**. Spotting it is the first move in almost every triangle question, so make sure you can point to it every time.

Tap the hypotenuse

An interactive activity.

Pythagoras' theorem 🔺

For any right-angled triangle with legs **a** and **b** and hypotenuse **c**: **a² + b² = c²** Use it whenever you know **two sides** and want the third, and **no angle** is involved. To find the hypotenuse: square both legs, add, then square-root.

Order the method

An interactive activity.

Find the hypotenuse

An interactive activity.

Another hypotenuse

An interactive activity.

Finding a shorter side ➖

If you know the hypotenuse and one leg, **subtract** instead: rearrange a² + b² = c² to **b² = c² − a²** For a hypotenuse of 13 and a leg of 5: b² = 13² − 5² = 169 − 25 = 144, so b = √144 = **12**. Subtract when the missing side is a leg, not the hypotenuse.

Find the missing leg

An interactive activity.

Which tool?

A right-angled triangle question gives you two sides and asks for the third, with NO angle mentioned. Which tool do you use?

• Pythagoras' theorem
• SOHCAHTOA (trigonometry)
• The area formula ½ base × height
• It does not matter which

Opposite, adjacent, hypotenuse 🏷️

For trig you label the sides **relative to the angle** you are using (call it θ): • **Hypotenuse** — opposite the right angle (longest), as before. • **Opposite** — the side across from θ, not touching it. • **Adjacent** — the side next to θ, between it and the right angle. Opposite and adjacent **swap** if you switch to the other angle — always label from the angle in the question.

Find the opposite side

You are using angle θ in a right-angled triangle. Which side is the OPPOSITE?

• The side across from θ that does not touch it
• The longest side
• The side next to θ, between it and the right angle
• Either of the two shorter sides

SOHCAHTOA ✳️

Three ratios connect an angle to two sides. Remember them with **SOH-CAH-TOA**: • **sin θ = Opposite ÷ Hypotenuse** • **cos θ = Adjacent ÷ Hypotenuse** • **tan θ = Opposite ÷ Adjacent** Pick the ratio that uses the two sides your question involves.

Match each rule to its formula

Finding a side with trig 📊

To find a missing side, choose the ratio with your angle and the two sides involved, then rearrange. With angle 30° and hypotenuse 10, to find the **opposite**: sin 30° = opposite ÷ 10, so **opposite = 10 × sin 30° = 10 × 0.5 = 5**. (sin 30° = 0.5 is one of the exact values worth knowing.)

Trig: find the side

An interactive activity.

Finding an angle

You know the opposite and adjacent sides and want the ANGLE. After writing tan θ = opposite ÷ adjacent, what do you do next?

• Apply inverse tan (tan⁻¹) to the result to get θ
• Multiply the ratio by the hypotenuse
• Use Pythagoras on the two sides
• Add the two sides together

The trek rules

Pythagoras' theorem says a² + b² = _____, where c is the hypotenuse. In a right-angled triangle, sin of an angle = opposite ÷ _____. To find an unknown ANGLE from two sides, use the _____ trig function, such as tan⁻¹.