The Law of Sines
The law of sines states that in any triangle, the ratio of each side to the sine of its opposite angle is the same. It solves triangles when you know two angles and a side, or two sides and a non-included angle, where the ambiguous case can give two valid triangles.
The Law of Sines, Example 2:
(75+42+x=180→ 117+x=180→x=180-117=63 ^circ )
The Law of Sines: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Unit-circle cosine
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Try one before moving on
The Law of Sines: pop-up practice
To find sides use the law of sines: (frac {a}{sin A}=frac {b}{sin B}=frac {c}{sin C})
(frac {22}{sin 75}=frac {b}{sin 42}= frac {c}{sin 63})
Now, use proportional ratios: (frac {a}{b}=frac{c}{d} → a×d=c×b)
(frac {22}{sin 75}=frac {b}{sin 42} → b=frac {22 × sin 42 } {sin 75} =frac{22 × 0.67}{0.96}=frac {14.74}{0.96}=15.35 cm)
(frac {22}{sin 75}= frac {c}{sin 63} → c=frac {22 × sin 63 } {sin 75} =frac{22 × 0.9}{0.96}=frac {19.8}{0.96}=20.62 cm)
Exercises for the Law of Sines
Find the side of c in the ABC triangle.
1.
2.
3.
- (color{blue}{73.33})
- (color{blue}{6.51})
- (color{blue}{20.53})
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