Study guides › Level 9 · Trigonometry

SOH-CAH-TOA: sin, cos and tan in a right triangle

Level 9 · Trigonometry · for questions like “Triangle ABC has a right angle at C. BC = 3, AC = 4 and AB = 5. What is sin A?”

💡 The idea

In a right triangle, the ratios between the sides depend only on the angle. Trigonometry gives them names. Everything starts with naming the sides from the angle you're looking at:

• Hypotenuse: the longest side, across from the square corner. It never changes.
• Opposite: the side across from your angle.
• Adjacent: the side next to your angle that isn't the hypotenuse.

🪜 The steps

1. Stand at the angle (say A) and name the three sides: opposite, adjacent, hypotenuse.
2. Use SOH-CAH-TOA:

- SOH: Sin = Opposite ÷ Hypotenuse

- CAH: Cos = Adjacent ÷ Hypotenuse

- TOA: Tan = Opposite ÷ Adjacent

3. Simplify the fraction if you can.

✏️ Worked examples

Triangle ABC with the square corner at C: BC = 3, AC = 4 and AB = 5.

1. From angle A: the opposite side is BC = 3, the adjacent side is AC = 4, and the hypotenuse is AB = 5.

sin A = 3/5, cos A = 4/5, tan A = 3/4.

2. From angle B, the sides swap roles: the opposite side is AC = 4 and the adjacent side is BC = 3.

sin B = 4/5, cos B = 3/5, tan B = 4/3.

Notice that sin A = cos B. That always happens with the two sharp angles of a right triangle.

⚠️ Watch out

• Opposite and adjacent swap when you move to the other angle. Always rename the sides first.
• The hypotenuse never goes in tan. Tan only uses opposite and adjacent.
• Sin and cos are never bigger than 1. If you get 5/3, you've divided the wrong way.

🎯 Try these

Triangle ABC, square corner at C: BC = 5, AC = 12 and AB = 13.

1. sin A = ?
2. cos A = ?
3. tan B = ?

Answers: 1) 5/13 2) 12/13 3) 12/5

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.