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 )

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.

1. (color{blue}{73.33})
2. (color{blue}{6.51})
3. (color{blue}{20.53})

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