**Find the Missing Sides and Angle Using Sin Law :**

The sine law is a relationship between the sides and angles in any triangle.

Let ABC be any triangle, where a, b, and c represent the measures of the sides opposite ∠A, ∠B, and ∠C, respectively. Then

a/sin A = b/sin B = c/sin C

(or)

sin A/a = sin B/b = sin C/c

**Question 1 :**

Determine the value of the marked unknown angle in each.

**Solution :**

(a) From the picture given above, we know that we need to find the angle C.

BC = a, AC = b = 31 m and AB = c = 28 m

<B = 62

a/sin A = b/sin B = c/sin C

31/sin 62 = 28/sin C

31/0.8829 = 28/sin C

35.11 = 28/sin C

sin C = 28/35.11

sin C = 0.7974

C = 52.88°

Hence the required angle is 53° (approximately).

(b) BC = a = 15, AC = b = 17.5 m and AB = c

<B = 98, <A = ?

a/sin A = b/sin B = c/sin C

15/sin A = 17.5/sin 98

15/sin A = 17.5/0.9902

15/sin A = 17.67

sin A = 15/17.67

sin A = 0.8488

Hence the required angle is 58° (approximately).

**Question 2 :**

Determining the lengths of all three sides and the measures of all three angles is called solving a triangle. Solve each triangle

**Solution :**

(a) AB = c = 13 m, BC = a, AC = b = 12 m

<B = 67

a/sin A = b/sin B = c/sin C ----(1)

a/sin A = 12/sin 67 = 13/sin C

12/sin 67 = 13/sin C

12/0.9205 = 13/sin C

13.03 = 13/sin C

sin C = 13/13.03

sin C = 0.9976

<C = 86

In triangle ABC,

<A + <B + <C = 180

<A + 67 + 86 = 180

<A + 153 = 180

<A = 180 - 153

<A = 27

a/sin 27 = 12/sin 67 = 13/sin 86

a/sin 27 = 12/sin 67

a/0.4539 = 13.03

a = 13.03(0.4539)

a = 5.91 approximately 6 m

Hence the missing side and missing angles are 6 m and 86 respectively.

(b) AB = c, BC = a, AC = b = 50 m

<A = 42, <B = 84

a/sin A = b/sin B = c/sin C

a/sin 42 = 50/sin 84 = c/sin C

a/sin 42 = 50/sin 84

a/0.6691 = 50/0.9945

a = 50.27 (0.6691)

a = 33.63 approximately 33.6 m

In triangle ABC,

<A + <B + <C = 180

42 + 84 + <C = 180

<C = 180 - 126

<C = 54

Hence the missing side and missing angle are 33.6 m and 54 degree.

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