Law of Sines worksheet for solving triangles, featuring a table with given sides and angles to fill in the missing values.

Let’s solve each problem step by step using the Law of Sines:

The Law of Sines says:
sin A / a = sin B / b = sin C / c

This means we can set up ratios between angles and their opposite sides to find missing parts.

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Problem 1:
Given:
a = 25 cm, b = 36 cm, A = 35°
Find: c, B, C

Step 1: Use Law of Sines to find angle B.
sin A / a = sin B / b
→ sin(35°) / 25 = sin B / 36

Calculate sin(35°) ≈ 0.5736
→ 0.5736 / 25 = sin B / 36
→ 0.022944 = sin B / 36
→ sin B = 0.022944 × 36 ≈ 0.825984

Now take inverse sine:
B ≈ arcsin(0.825984) ≈ 55.7° (let’s round to 56° for simplicity, but we’ll keep decimal for accuracy)

Actually, let’s use more precise calculation:
arcsin(0.825984) ≈ 55.68° → B ≈ 55.7°

Step 2: Find angle C.
Sum of angles in triangle = 180°
C = 180° - A - B = 180 - 35 - 55.7 = 89.3°

Step 3: Find side c.
Use Law of Sines again:
sin A / a = sin C / c
→ sin(35°)/25 = sin(89.3°)/c

sin(89.3°) ≈ 0.9999 (almost 1)
→ 0.5736 / 25 = 0.9999 / c
→ 0.022944 = 0.9999 / c
→ c = 0.9999 / 0.022944 ≈ 43.6 cm

So Problem 1:
c ≈ 43.6 cm, B ≈ 55.7°, C ≈ 89.3°

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Problem 2:
Given:
b = 25 cm, A = 42°, B = 75°
Find: a, c, C

Step 1: Find angle C.
C = 180 - A - B = 180 - 42 - 75 = 63°

Step 2: Find side a.
Use Law of Sines:
sin A / a = sin B / b
→ sin(42°)/a = sin(75°)/25

sin(42°) ≈ 0.6691, sin(75°) ≈ 0.9659
→ 0.6691 / a = 0.9659 / 25
→ 0.6691 / a = 0.038636
→ a = 0.6691 / 0.038636 ≈ 17.3 cm

Step 3: Find side c.
sin B / b = sin C / c
→ 0.9659 / 25 = sin(63°)/c
sin(63°) ≈ 0.8910
→ 0.038636 = 0.8910 / c
→ c = 0.8910 / 0.038636 ≈ 23.1 cm

So Problem 2:
a ≈ 17.3 cm, c ≈ 23.1 cm, C = 63°

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Problem 3:
Given:
a = 42 cm, b = 25 cm, A = 38°
Find: c, B, C

Step 1: Find angle B.
sin A / a = sin B / b
→ sin(38°)/42 = sin B / 25

sin(38°) ≈ 0.6157
→ 0.6157 / 42 = sin B / 25
→ 0.01466 = sin B / 25
→ sin B = 0.01466 × 25 ≈ 0.3665

B = arcsin(0.3665) ≈ 21.5°

Step 2: Find angle C.
C = 180 - 38 - 21.5 = 120.5°

Step 3: Find side c.
sin A / a = sin C / c
→ 0.6157 / 42 = sin(120.5°)/c

sin(120.5°) = sin(180 - 120.5) = sin(59.5°) ≈ 0.8616
→ 0.01466 = 0.8616 / c
→ c = 0.8616 / 0.01466 ≈ 58.8 cm

So Problem 3:
c ≈ 58.8 cm, B ≈ 21.5°, C ≈ 120.5°

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Problem 4:
Given:
a = 9.2 m, b = 4.6 m, c = 5.8 m
Find: A, B, C

We’ll use Law of Cosines first because we have all three sides.

Law of Cosines:
cos A = (b² + c² - a²) / (2bc)
cos B = (a² + c² - b²) / (2ac)
cos C = (a² + b² - c²) / (2ab)

Step 1: Find angle A.
cos A = (4.6² + 5.8² - 9.2²) / (2 × 4.6 × 5.8)
= (21.16 + 33.64 - 84.64) / (53.36)
= (-29.84) / 53.36 ≈ -0.5592
A = arccos(-0.5592) ≈ 124.0°

Step 2: Find angle B.
cos B = (9.2² + 5.8² - 4.6²) / (2 × 9.2 × 5.8)
= (84.64 + 33.64 - 21.16) / (106.72)
= 97.12 / 106.72 ≈ 0.9100
B = arccos(0.9100) ≈ 24.5°

Step 3: Find angle C.
C = 180 - A - B = 180 - 124.0 - 24.5 = 31.5°

Check with Law of Cosines for C:
cos C = (9.2² + 4.6² - 5.8²) / (2 × 9.2 × 4.6)
= (84.64 + 21.16 - 33.64) / (84.64)
= 72.16 / 84.64 ≈ 0.8525
C = arccos(0.8525) ≈ 31.5° ✔ matches!

So Problem 4:
A ≈ 124.0°, B ≈ 24.5°, C ≈ 31.5°

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Final Answer:
1. c ≈ 43.6 cm, B ≈ 55.7°, C ≈ 89.3°
2. a ≈ 17.3 cm, c ≈ 23.1 cm, C = 63°
3. c ≈ 58.8 cm, B ≈ 21.5°, C ≈ 120.5°
4. A ≈ 124.0°, B ≈ 24.5°, C ≈ 31.5°