Lesson · Grade 11 · Pre Calculus
The Sine Law: How Far Away Is the Fire?
- Grade 11
- Pre Calculus
- 2 min 13 s
- British Columbia
- BC11-MA-79
Two lookout towers 12 km apart measure angles of 65° and 75° to the same fire. The video finds the third angle (40°) and uses a/sin A = b/sin B = c/sin C to show the fire is about 16.9 km from one tower and 18.0 km from the other. It colours each side to match its opposite angle, warns against pairing a side with an angle that touches it, checks the answer by matching the smallest angle to the shortest side and the largest to the longest, and ends on a one-line recap of the sine law.
Concept: The sine law: solving a triangle with no right angle
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In one sentence
The question this lesson answers
Two lookout towers, B and C, are 12 kilometres apart. Both spot a fire at A.
Tower B measures 65 degrees, and tower C measures 75 degrees.
Transcript
What the lesson says
Two lookout towers, B and C, are 12 kilometres apart. Both spot a fire at A.
Tower B measures 65 degrees, and tower C measures 75 degrees.
How far is the fire from each tower? With no right angle, we use the sine law.
The side opposite an angle is the one it does not touch: side a is opposite angle A, b is opposite B, and c is opposite C.
The sine law says a over sine A equals b over sine B equals c over sine C.
To use it, we need one side and its opposite angle. We know side a, but not angle A.
The angles of a triangle add to 180 degrees, so A is 180 minus 65 minus 75, or 40 degrees.
Now a and A are a known pair. To find b, write 12 over sine 40 degrees equals b over sine 65 degrees.
Multiply both sides by sine 65 degrees: b equals 12 times sine 65 degrees over sine 40 degrees.
In degree mode on a calculator, that is 12 times 0.906 over 0.643, about 16.9 kilometres.
In the same way, c equals 12 times sine 75 degrees over sine 40 degrees, about 18.0 kilometres.
A common mistake is pairing side a with angle B or C. Those touch side a. Its opposite angle is A.
Check: the smallest angle faces the shortest side, and the largest faces the longest.
So the fire is about 16.9 kilometres from tower C and 18.0 kilometres from tower B.
The sine law: each side over the sine of its opposite angle gives the same number.
Curriculum
Where this fits
- British Columbia, Grade 11
Pre Calculus 11, Trigonometry: trigonometry: non-right triangles and angles in standard positionBC11-MA-79
The outcomes and strands follow the British Columbia curriculum.
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