Sine rule
Apply the sine rule to find unknown sides and angles in non-right-angled triangles; recognise and solve the ambiguous case where two triangles are possible.
Worked examples
Finding an unknown side using the sine rule
Straightforward
Problem
In triangle , angle , angle , and side cm. Find side .
1
Write down the sine rule and identify the known and unknown quantities.
Sine rule:
Known: cm, , . Find: .
Known: cm, , . Find: .
2
Substitute the known values.
3
Solve for .
Answer
Side cm.
Finding an unknown angle using the sine rule
Moderate
Problem
In triangle , cm, cm, and angle . Find angle (assuming it is acute).
1
Write the sine rule rearranged to find .
2
Substitute the known values.
3
Since the question states is acute, we take this first-quadrant value.
Find angle using the inverse sine function.
Since the question states is acute, we take this first-quadrant value.
Answer
Angle .
The ambiguous case — two possible triangles
Challenging
Problem
In triangle , cm, cm, and angle . Show that two triangles are possible and find both possible values of angle .
1
Use the sine rule to find .
2
Check: ✓
So — a valid triangle.
Find the first possible angle .
Check: ✓
So — a valid triangle.
3
Find the second possible angle (the ambiguous case).
Since , a second solution exists:
Check: ✓
So — also a valid triangle.
Check: ✓
So — also a valid triangle.
4
State the conclusion.
Both triangles are geometrically valid. Without additional information (e.g. a diagram or the size of angle ), we cannot determine which triangle is the intended one.
Answer
Two triangles are possible:
- Triangle 1: ,
- Triangle 2: ,
- Triangle 1: ,
- Triangle 2: ,
Practise
Q1·Straightforward
In triangle , angle , angle , and side cm (the side opposite angle ). Use the sine rule to find side (opposite angle ). Give your answer to 2 decimal places.
Explanation
Using the sine rule:
Q2·Straightforward
In triangle , angle , angle , and side cm (opposite angle ). Find side (opposite angle ) using the sine rule. Give your answer to 2 decimal places.
Explanation
Using the sine rule:
Q3·Straightforward
In triangle , angle , angle , and side cm (opposite angle ). Find side (opposite angle ). Give your answer to 2 decimal places.
Explanation
Using the sine rule:
Q4·Straightforward
In triangle , angle , angle , and m (opposite angle ). Find side (opposite angle ). Give your answer to 2 decimal places.
Explanation
Using the sine rule:
Q5·Moderate
In triangle , cm, cm, and angle (opposite side ). Assuming angle is acute, find angle to 1 decimal place.
Explanation
Rearranging the sine rule:
Since the question states is acute, we take the first-quadrant value.
Since the question states is acute, we take the first-quadrant value.
Q6·Moderate
In triangle , cm, cm, and angle (opposite side ). Assuming angle is acute, find angle to 1 decimal place.
Explanation
Using the sine rule:
Q7·Moderate
In triangle , angle , angle , and cm (opposite angle ). Find side (opposite angle ). Give your answer to 1 decimal place.
Explanation
Find angle :
Apply the sine rule:
Apply the sine rule:
Q8·Moderate
Two park rangers, and , are stationed km apart. They both spot a fire . Ranger measures the angle and ranger measures angle . Find the distance from ranger to the fire () to 2 decimal places.
Explanation
In triangle :
Apply the sine rule (the side km is opposite angle ; side is opposite angle ):
Apply the sine rule (the side km is opposite angle ; side is opposite angle ):
Q9·Moderate
In triangle , angle , angle , and m (the side opposite angle ). Find side (opposite angle ) to 2 decimal places.
Explanation
Find angle :
Apply the sine rule:
Apply the sine rule:
Q10·Challenging
In triangle , cm, cm, and angle (opposite side ). Two triangles are possible (the ambiguous case). Find the **larger** of the two possible values of angle , to the nearest degree.
Explanation
Using the sine rule:
First solution:
Check: ✓ → (valid triangle)
Second solution (ambiguous case):
Check: ✓ → (valid triangle)
Both triangles are possible. The larger value of angle is .
First solution:
Check: ✓ → (valid triangle)
Second solution (ambiguous case):
Check: ✓ → (valid triangle)
Both triangles are possible. The larger value of angle is .
Q11·Challenging
A triangular block of land has vertices , and . Side m, angle , and angle . Find the length of side to the nearest metre.
Explanation
Find angle :
Side m is opposite angle . Side is opposite angle .
Apply the sine rule:
Side m is opposite angle . Side is opposite angle .
Apply the sine rule:
Q12·Challenging
Points and are on opposite sides of a river. A point is on the same bank as , with m. From , the angle ; from , the angle . Find the width of the river to the nearest metre.
Explanation
In triangle :
Side m is opposite angle . The river width is opposite angle .
Apply the sine rule:
Side m is opposite angle . The river width is opposite angle .
Apply the sine rule:
Open Math
Sine rule
Measurement · MS-M6
Name:
Date:
Q1Straightforward
In triangle , angle , angle , and side cm (the side opposite angle ). Use the sine rule to find side (opposite angle ). Give your answer to 2 decimal places.
Q2Straightforward
In triangle , angle , angle , and side cm (opposite angle ). Find side (opposite angle ) using the sine rule. Give your answer to 2 decimal places.
Q3Straightforward
In triangle , angle , angle , and side cm (opposite angle ). Find side (opposite angle ). Give your answer to 2 decimal places.
Q4Straightforward
In triangle , angle , angle , and m (opposite angle ). Find side (opposite angle ). Give your answer to 2 decimal places.
Q5Moderate
In triangle , cm, cm, and angle (opposite side ). Assuming angle is acute, find angle to 1 decimal place.
Q6Moderate
In triangle , cm, cm, and angle (opposite side ). Assuming angle is acute, find angle to 1 decimal place.
Q7Moderate
In triangle , angle , angle , and cm (opposite angle ). Find side (opposite angle ). Give your answer to 1 decimal place.
Q8Moderate
Two park rangers, and , are stationed km apart. They both spot a fire . Ranger measures the angle and ranger measures angle . Find the distance from ranger to the fire () to 2 decimal places.
Q9Moderate
In triangle , angle , angle , and m (the side opposite angle ). Find side (opposite angle ) to 2 decimal places.
Q10Challenging
In triangle , cm, cm, and angle (opposite side ). Two triangles are possible (the ambiguous case). Find the **larger** of the two possible values of angle , to the nearest degree.
Q11Challenging
A triangular block of land has vertices , and . Side m, angle , and angle . Find the length of side to the nearest metre.
Q12Challenging
Points and are on opposite sides of a river. A point is on the same bank as , with m. From , the angle ; from , the angle . Find the width of the river to the nearest metre.
Worked solutions and answers at openmath.au/year-12/standard-2/non-right-angled-trigonometry/sine-rule