📘 Remember
- Label the triangle before choosing a formula.
- Use degree mode unless the question says otherwise.
- Keep full calculator values until the final rounding step.
🎯 Learning Objectives
- use inverse sine, cosine and tangent;
- select the correct inverse ratio;
- use a calculator in degree mode;
- check that right-triangle angles are acute.
Introduction
This lesson develops Finding Missing Angles from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Inverse Trigonometry
When the unknown is an angle, use sin⁻¹, cos⁻¹ or tan⁻¹ after forming the ratio.
Worked Example
tan θ = 6 ÷ 9
θ = tan⁻¹(6 ÷ 9)
θ ≈ 33.7°
Calculator Reminder
Use degree mode. The inverse function usually requires SHIFT or 2nd followed by sin, cos or tan.
Practice Questions
- Opposite=5, hypotenuse=13. Find θ.
- Adjacent=8, hypotenuse=10. Find θ.
- Opposite=12, adjacent=7. Find θ.
- Explain why an answer of 126° cannot be an acute angle in a right triangle.
✅ Check Your Work
Show answers and working
- θ=sin⁻¹(5/13)≈22.6°.
- θ=cos⁻¹(8/10)≈36.9°.
- θ=tan⁻¹(12/7)≈59.7°.
- The two non-right angles in a right triangle must both be less than 90°.
🧭 Interactive Lab: Inverse-Trig Angle Finder
Change the values or make a choice and watch the mathematics respond.
Summary
⚠️ Common Mistakes
- Using the wrong reference angle.
- Rounding too early.
- Forgetting units or the degree symbol.
⭐ Exam Tips
- Write the formula before substituting.
- Sketch and label any diagram that is not already provided.
- Check whether the answer is sensible for the shape.
🌍 Did You Know?
Surveyors, engineers, pilots and architects use trigonometry to calculate distances that are difficult or unsafe to measure directly.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.