TRIGONOMETRY · TEXTBOOK LESSON
Trigonometry connects angles and side lengths. In right-angled triangles, sine, cosine and tangent are ratios; in more general triangles, the sine and cosine rules extend these relationships.
Learning objectives
By the end of this lesson, you should be able to:
- Identify opposite, adjacent and hypotenuse correctly.
- Select sine, cosine or tangent for right-angled triangle problems.
- Rearrange trigonometric equations to find missing sides or angles.
- Check calculator mode and interpret answers with suitable accuracy.
1. Key vocabulary and definition
Definition
For an angle θ in a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
hypotenuseoppositeadjacentsinecosinetangent
2. Essential facts and rules
These are the facts you should know before attempting the worked example.
Key mathematical facts
- The hypotenuse is always opposite the right angle.
- SOH-CAH-TOA is a memory aid for the three basic ratios.
- Use inverse trig functions to find an angle from a ratio.
3. Standard method
A reliable method helps prevent errors and makes your reasoning easy to follow.
1
Mark the right angle and the angle of interest.
2
Label the sides H, O and A relative to that angle.
3
Choose the ratio containing the known and unknown sides.
4
Substitute values and rearrange.
5
Calculate and round only at the end.
4. Worked example
Worked example
In a right triangle, the hypotenuse is 10 cm and an angle is 35°. Find the side opposite the angle.
1
Opposite and hypotenuse are involved, so use sine.
2
sin 35° = opposite / 10.
3
opposite = 10 sin 35°.
4
opposite ≈ 5.74 cm.
5. Common mistakes
Watch out for these
- Starting calculations before deciding what the question is asking.
- Skipping important steps or changing notation midway through a solution.
- Accepting an answer without checking whether its size, sign or unit is sensible.
6. Video lesson
This topic is ready for a matching video lesson. The written textbook lesson is fully available now; additional videos can be added to the same lesson template.
7. Practice exercise
Complete these without looking at the answers first.
- State sin θ in terms of O, A and H.
- A right triangle has H=13 and A=5. Find O.
- Find x if tan 40° = x/12.
- Find θ if sin θ = 0.6.
- Explain why calculator degree/radian mode matters.
Show answers
- O/H
- 12
- ≈10.07
- ≈36.9°
- The same numerical input represents different angle measures in degree and radian modes
8. Chapter summary
- Understand the underlying idea before memorising a procedure.
- Use precise mathematical notation and show a logical method.
- Always check the final result.