Ignoring the ambiguous case H
"The Missing Answer"
The Mistake in Action
In triangle ABC, a = 10cm, b = 15cm, and angle A = 40°. Find angle B.
Incomplete: $\frac{\sin B}{15} = \frac{\sin 40°}{10}$ $\sin B = \frac{15 \times \sin 40°}{10} = 0.964$ $B = \sin^{-1}(0.964) = 74.6°$
Why It Happens
Students find one answer and stop, forgetting that $\sin \theta = \sin(180° - \theta)$.
The Fix
When finding an angle using the sine rule, check for two possible answers.
$\sin B = 0.964$
First possibility: $B = 74.6°$ Second possibility: $B = 180° - 74.6° = 105.4°$
Check both work:
- If $B = 74.6°$: $A + B = 40° + 74.6° = 114.6°$ ✓ (leaves room for angle C)
- If $B = 105.4°$: $A + B = 40° + 105.4° = 145.4°$ ✓ (leaves $34.6°$ for angle C)
Both are valid! The triangle could have either shape.
When is there only one answer?
- When the second answer would make angles sum to more than 180°
- When you're finding an angle you know is acute or obtuse
Spot the Mistake
Can you identify where this student went wrong?
$\sin B = 0.964$
$B = 74.6°$
Click on the line that contains the error.
Related Topics
Learn more about the underlying maths: