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AQA GCSE · Question 19 · Geometry and Measures
Use the sine rule to work out length x. You must show your working.
Use the sine rule to work out length x. You must show your working.
How to approach this question
1. The sine rule requires pairs of sides and their opposite angles. You have side 54 cm, but you don't have its opposite angle. Calculate this missing angle first.
2. The sum of angles in a triangle is 180°.
3. Once you have the angle opposite the 54 cm side, you can set up the sine rule equation: x / sin(A) = b / sin(B).
4. Substitute the known values into the equation. The side you want to find is x, and its opposite angle is 35°.
5. Rearrange the equation to make x the subject and calculate the final value.
Full Answer
1. **Find the third angle in the triangle.**
The sum of angles in a triangle is 180°.
Third angle = 180° - 117° - 35° = 28°.
2. **Set up the sine rule.**
The sine rule is a/sin(A) = b/sin(B).
We have a side-angle pair (54 cm and 28°) and we want to find side x which is opposite the 35° angle.
x / sin(35°) = 54 / sin(28°)
3. **Rearrange and solve for x.**
x = (54 / sin(28°)) * sin(35°)
x = (54 / 0.4694...) * 0.5735...
x = 115.01... * 0.5735...
x = 65.93...
So, x ≈ 65.9 cm (to 1 decimal place).
The sine rule states that for any triangle with sides a, b, c and opposite angles A, B, C respectively:
a/sin(A) = b/sin(B) = c/sin(C)
Step 1: Find the missing angle.
To use the sine rule, we need a known side and its opposite angle. We have the side 54 cm, so we need the angle opposite it.
The sum of angles in a triangle is 180°.
Missing angle = 180° - 117° - 35° = 28°.
Step 2: Apply the sine rule.
Now we can set up the equation. We pair the side we want to find (x) with its opposite angle (35°), and the known side (54 cm) with its opposite angle (28°).
x / sin(35°) = 54 / sin(28°)
Step 3: Solve for x.
To isolate x, we multiply both sides by sin(35°):
x = (54 * sin(35°)) / sin(28°)
Using a calculator:
x = (54 * 0.573576...) / 0.469471...
x = 30.973... / 0.469471...
x = 65.973... cm
Rounding to one decimal place, x = 65.9 cm.
Common mistakes
✗ Trying to use SOH CAH TOA, which only works for right-angled triangles.
✗ Mismatching sides and angles in the sine rule formula, e.g., x / sin(28°) = 54 / sin(35°).
✗ Forgetting to find the third angle and trying to use 117° in the formula.
✗ Calculator mode being in radians instead of degrees.
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