Cosine Rule — Step 1 — Part 2 — Dynamic Practice
Cosine Rule — Step 1 — Part 2 — Dynamic Practice. Practice questions to deepen understanding of using the cosine rule to find a missing side — advanced practice. Online math practice with full solutions and clear explanations.
Dynamic advanced practice in the cosine rule (step 1, Part 2) — using c² = a² + b² − 2ab·cos C to find a missing side in more varied configurations. New questions every attempt.
Question 1
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 8
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 8
• side b = 8
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 8^2 + 8^2 - 2 \cdot 8 \cdot 8 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 64 + 64 - 2 \cdot 8 \cdot 8 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 17.15\)
\(c = \sqrt{17.15} = 4.14\)
\(c^2 = 64 + 64 - 2 \cdot 8 \cdot 8 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 17.15\)
\(c = \sqrt{17.15} = 4.14\)
Answer: c = 4.14
Question 2
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 9
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 5
• side b = 9
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 5^2 + 9^2 - 2 \cdot 5 \cdot 9 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 25 + 81 - 2 \cdot 5 \cdot 9 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 42.36\)
\(c = \sqrt{42.36} = 6.51\)
\(c^2 = 25 + 81 - 2 \cdot 5 \cdot 9 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 42.36\)
\(c = \sqrt{42.36} = 6.51\)
Answer: c = 6.51
Question 3
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 4
• side b = 6
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 4
• side b = 6
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 4^2 + 6^2 - 2 \cdot 4 \cdot 6 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 16 + 36 - 2 \cdot 4 \cdot 6 \cdot \frac{1}{2}\)
\(c^2 = 28\)
\(c = \sqrt{28} = 5.29\)
\(c^2 = 16 + 36 - 2 \cdot 4 \cdot 6 \cdot \frac{1}{2}\)
\(c^2 = 28\)
\(c = \sqrt{28} = 5.29\)
Answer: c = 5.29
Question 4
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 10
• side b = 4
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 10
• side b = 4
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 10^2 + 4^2 - 2 \cdot 10 \cdot 4 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 100 + 16 - 2 \cdot 10 \cdot 4 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 59.43\)
\(c = \sqrt{59.43} = 7.71\)
\(c^2 = 100 + 16 - 2 \cdot 10 \cdot 4 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 59.43\)
\(c = \sqrt{59.43} = 7.71\)
Answer: c = 7.71
Question 5
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 11
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 5
• side b = 11
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 5^2 + 11^2 - 2 \cdot 5 \cdot 11 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 25 + 121 - 2 \cdot 5 \cdot 11 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 50.74\)
\(c = \sqrt{50.74} = 7.12\)
\(c^2 = 25 + 121 - 2 \cdot 5 \cdot 11 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 50.74\)
\(c = \sqrt{50.74} = 7.12\)
Answer: c = 7.12
Question 6
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 8
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 8
• side b = 8
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 8^2 + 8^2 - 2 \cdot 8 \cdot 8 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 64 + 64 - 2 \cdot 8 \cdot 8 \cdot \frac{1}{2}\)
\(c^2 = 64\)
\(c = \sqrt{64} = 8\)
\(c^2 = 64 + 64 - 2 \cdot 8 \cdot 8 \cdot \frac{1}{2}\)
\(c^2 = 64\)
\(c = \sqrt{64} = 8\)
Answer: c = 8
Question 7
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 7
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 8
• side b = 7
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 8^2 + 7^2 - 2 \cdot 8 \cdot 7 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 64 + 49 - 2 \cdot 8 \cdot 7 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 16.01\)
\(c = \sqrt{16.01} = 4\)
\(c^2 = 64 + 49 - 2 \cdot 8 \cdot 7 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 16.01\)
\(c = \sqrt{16.01} = 4\)
Answer: c = 4
Question 8
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 6
• side b = 5
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 6
• side b = 5
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 6^2 + 5^2 - 2 \cdot 6 \cdot 5 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 36 + 25 - 2 \cdot 6 \cdot 5 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 18.57\)
\(c = \sqrt{18.57} = 4.31\)
\(c^2 = 36 + 25 - 2 \cdot 6 \cdot 5 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 18.57\)
\(c = \sqrt{18.57} = 4.31\)
Answer: c = 4.31
Question 9
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 6
• side b = 11
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 6
• side b = 11
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 6^2 + 11^2 - 2 \cdot 6 \cdot 11 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 36 + 121 - 2 \cdot 6 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 91\)
\(c = \sqrt{91} = 9.54\)
\(c^2 = 36 + 121 - 2 \cdot 6 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 91\)
\(c = \sqrt{91} = 9.54\)
Answer: c = 9.54
Question 10
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 7
• side b = 10
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 7
• side b = 10
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 7^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 49 + 100 - 2 \cdot 7 \cdot 10 \cdot \frac{1}{2}\)
\(c^2 = 79\)
\(c = \sqrt{79} = 8.89\)
\(c^2 = 49 + 100 - 2 \cdot 7 \cdot 10 \cdot \frac{1}{2}\)
\(c^2 = 79\)
\(c = \sqrt{79} = 8.89\)
Answer: c = 8.89
Question 11
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 9
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 5
• side b = 9
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 5^2 + 9^2 - 2 \cdot 5 \cdot 9 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 25 + 81 - 2 \cdot 5 \cdot 9 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 28.06\)
\(c = \sqrt{28.06} = 5.3\)
\(c^2 = 25 + 81 - 2 \cdot 5 \cdot 9 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 28.06\)
\(c = \sqrt{28.06} = 5.3\)
Answer: c = 5.3
Question 12
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 7
• side b = 7
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 7
• side b = 7
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 7^2 + 7^2 - 2 \cdot 7 \cdot 7 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 49 + 49 - 2 \cdot 7 \cdot 7 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 13.13\)
\(c = \sqrt{13.13} = 3.62\)
\(c^2 = 49 + 49 - 2 \cdot 7 \cdot 7 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 13.13\)
\(c = \sqrt{13.13} = 3.62\)
Answer: c = 3.62
Question 13
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 4
• side b = 8
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 4
• side b = 8
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 4^2 + 8^2 - 2 \cdot 4 \cdot 8 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 16 + 64 - 2 \cdot 4 \cdot 8 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 34.75\)
\(c = \sqrt{34.75} = 5.89\)
\(c^2 = 16 + 64 - 2 \cdot 4 \cdot 8 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 34.75\)
\(c = \sqrt{34.75} = 5.89\)
Answer: c = 5.89
Question 14
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 11
• side b = 5
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 11
• side b = 5
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 11^2 + 5^2 - 2 \cdot 11 \cdot 5 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 121 + 25 - 2 \cdot 11 \cdot 5 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 50.74\)
\(c = \sqrt{50.74} = 7.12\)
\(c^2 = 121 + 25 - 2 \cdot 11 \cdot 5 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 50.74\)
\(c = \sqrt{50.74} = 7.12\)
Answer: c = 7.12
Question 15
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 11
• side b = 9
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 11
• side b = 9
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 11^2 + 9^2 - 2 \cdot 11 \cdot 9 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 121 + 81 - 2 \cdot 11 \cdot 9 \cdot \frac{1}{2}\)
\(c^2 = 103\)
\(c = \sqrt{103} = 10.15\)
\(c^2 = 121 + 81 - 2 \cdot 11 \cdot 9 \cdot \frac{1}{2}\)
\(c^2 = 103\)
\(c = \sqrt{103} = 10.15\)
Answer: c = 10.15
Question 16
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 4
• side b = 10
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 4
• side b = 10
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 4^2 + 10^2 - 2 \cdot 4 \cdot 10 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 16 + 100 - 2 \cdot 4 \cdot 10 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 46.72\)
\(c = \sqrt{46.72} = 6.84\)
\(c^2 = 16 + 100 - 2 \cdot 4 \cdot 10 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 46.72\)
\(c = \sqrt{46.72} = 6.84\)
Answer: c = 6.84
Question 17
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 9
• side b = 5
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 9
• side b = 5
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 9^2 + 5^2 - 2 \cdot 9 \cdot 5 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 81 + 25 - 2 \cdot 9 \cdot 5 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 42.36\)
\(c = \sqrt{42.36} = 6.51\)
\(c^2 = 81 + 25 - 2 \cdot 9 \cdot 5 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 42.36\)
\(c = \sqrt{42.36} = 6.51\)
Answer: c = 6.51
Question 18
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 6
• side b = 6
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 6
• side b = 6
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 6^2 + 6^2 - 2 \cdot 6 \cdot 6 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 36 + 36 - 2 \cdot 6 \cdot 6 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 21.09\)
\(c = \sqrt{21.09} = 4.59\)
\(c^2 = 36 + 36 - 2 \cdot 6 \cdot 6 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 21.09\)
\(c = \sqrt{21.09} = 4.59\)
Answer: c = 4.59
Question 19
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 10
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 8
• side b = 10
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 8^2 + 10^2 - 2 \cdot 8 \cdot 10 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 64 + 100 - 2 \cdot 8 \cdot 10 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 50.86\)
\(c = \sqrt{50.86} = 7.13\)
\(c^2 = 64 + 100 - 2 \cdot 8 \cdot 10 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 50.86\)
\(c = \sqrt{50.86} = 7.13\)
Answer: c = 7.13
Question 20
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 7
• side b = 11
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 7
• side b = 11
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 7^2 + 11^2 - 2 \cdot 7 \cdot 11 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 49 + 121 - 2 \cdot 7 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 61.11\)
\(c = \sqrt{61.11} = 7.82\)
\(c^2 = 49 + 121 - 2 \cdot 7 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 61.11\)
\(c = \sqrt{61.11} = 7.82\)
Answer: c = 7.82
Question 21
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 4
• side b = 5
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 4
• side b = 5
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 4^2 + 5^2 - 2 \cdot 4 \cdot 5 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 16 + 25 - 2 \cdot 4 \cdot 5 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 6.36\)
\(c = \sqrt{6.36} = 2.52\)
\(c^2 = 16 + 25 - 2 \cdot 4 \cdot 5 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 6.36\)
\(c = \sqrt{6.36} = 2.52\)
Answer: c = 2.52
Question 22
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 11
• side b = 4
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 11
• side b = 4
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 11^2 + 4^2 - 2 \cdot 11 \cdot 4 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 121 + 16 - 2 \cdot 11 \cdot 4 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 74.77\)
\(c = \sqrt{74.77} = 8.65\)
\(c^2 = 121 + 16 - 2 \cdot 11 \cdot 4 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 74.77\)
\(c = \sqrt{74.77} = 8.65\)
Answer: c = 8.65
Question 23
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 11
• side b = 6
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 11
• side b = 6
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 11^2 + 6^2 - 2 \cdot 11 \cdot 6 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 121 + 36 - 2 \cdot 11 \cdot 6 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 42.68\)
\(c = \sqrt{42.68} = 6.53\)
\(c^2 = 121 + 36 - 2 \cdot 11 \cdot 6 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 42.68\)
\(c = \sqrt{42.68} = 6.53\)
Answer: c = 6.53
Question 24
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 9
• side b = 6
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 9
• side b = 6
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 9^2 + 6^2 - 2 \cdot 9 \cdot 6 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 81 + 36 - 2 \cdot 9 \cdot 6 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 40.63\)
\(c = \sqrt{40.63} = 6.37\)
\(c^2 = 81 + 36 - 2 \cdot 9 \cdot 6 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 40.63\)
\(c = \sqrt{40.63} = 6.37\)
Answer: c = 6.37
Question 25
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 11
• side b = 11
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 11
• side b = 11
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 11^2 + 11^2 - 2 \cdot 11 \cdot 11 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 121 + 121 - 2 \cdot 11 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 121\)
\(c = \sqrt{121} = 11\)
\(c^2 = 121 + 121 - 2 \cdot 11 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 121\)
\(c = \sqrt{121} = 11\)
Answer: c = 11
Question 26
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 7
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 5
• side b = 7
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 25 + 49 - 2 \cdot 5 \cdot 7 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 24.5\)
\(c = \sqrt{24.5} = 4.95\)
\(c^2 = 25 + 49 - 2 \cdot 5 \cdot 7 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 24.5\)
\(c = \sqrt{24.5} = 4.95\)
Answer: c = 4.95
Question 27
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 10
• side b = 4
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 10
• side b = 4
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 10^2 + 4^2 - 2 \cdot 10 \cdot 4 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 100 + 16 - 2 \cdot 10 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 76\)
\(c = \sqrt{76} = 8.72\)
\(c^2 = 100 + 16 - 2 \cdot 10 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 76\)
\(c = \sqrt{76} = 8.72\)
Answer: c = 8.72
Question 28
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 10
• side b = 9
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 10
• side b = 9
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 10^2 + 9^2 - 2 \cdot 10 \cdot 9 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 100 + 81 - 2 \cdot 10 \cdot 9 \cdot \frac{1}{2}\)
\(c^2 = 91\)
\(c = \sqrt{91} = 9.54\)
\(c^2 = 100 + 81 - 2 \cdot 10 \cdot 9 \cdot \frac{1}{2}\)
\(c^2 = 91\)
\(c = \sqrt{91} = 9.54\)
Answer: c = 9.54
Question 29
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 6
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 5
• side b = 6
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 5^2 + 6^2 - 2 \cdot 5 \cdot 6 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 25 + 36 - 2 \cdot 5 \cdot 6 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 18.57\)
\(c = \sqrt{18.57} = 4.31\)
\(c^2 = 25 + 36 - 2 \cdot 5 \cdot 6 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 18.57\)
\(c = \sqrt{18.57} = 4.31\)
Answer: c = 4.31
Question 30
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 11
• side b = 10
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 11
• side b = 10
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 11^2 + 10^2 - 2 \cdot 11 \cdot 10 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 121 + 100 - 2 \cdot 11 \cdot 10 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 65.44\)
\(c = \sqrt{65.44} = 8.09\)
\(c^2 = 121 + 100 - 2 \cdot 11 \cdot 10 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 65.44\)
\(c = \sqrt{65.44} = 8.09\)
Answer: c = 8.09
Question 31
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 11
• side b = 8
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 11
• side b = 8
• Angle C (between the sides) = 30°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 11^2 + 8^2 - 2 \cdot 11 \cdot 8 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 121 + 64 - 2 \cdot 11 \cdot 8 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 32.58\)
\(c = \sqrt{32.58} = 5.71\)
\(c^2 = 121 + 64 - 2 \cdot 11 \cdot 8 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 32.58\)
\(c = \sqrt{32.58} = 5.71\)
Answer: c = 5.71
Question 32
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 10
• side b = 9
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 10
• side b = 9
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 10^2 + 9^2 - 2 \cdot 10 \cdot 9 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 100 + 81 - 2 \cdot 10 \cdot 9 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 53.72\)
\(c = \sqrt{53.72} = 7.33\)
\(c^2 = 100 + 81 - 2 \cdot 10 \cdot 9 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 53.72\)
\(c = \sqrt{53.72} = 7.33\)
Answer: c = 7.33
Question 33
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 8
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 5
• side b = 8
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 5^2 + 8^2 - 2 \cdot 5 \cdot 8 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 25 + 64 - 2 \cdot 5 \cdot 8 \cdot \frac{1}{2}\)
\(c^2 = 49\)
\(c = \sqrt{49} = 7\)
\(c^2 = 25 + 64 - 2 \cdot 5 \cdot 8 \cdot \frac{1}{2}\)
\(c^2 = 49\)
\(c = \sqrt{49} = 7\)
Answer: c = 7
Question 34
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 11
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 8
• side b = 11
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 8^2 + 11^2 - 2 \cdot 8 \cdot 11 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 64 + 121 - 2 \cdot 8 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 60.55\)
\(c = \sqrt{60.55} = 7.78\)
\(c^2 = 64 + 121 - 2 \cdot 8 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 60.55\)
\(c = \sqrt{60.55} = 7.78\)
Answer: c = 7.78
Question 35
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 11
• side b = 5
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 11
• side b = 5
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 11^2 + 5^2 - 2 \cdot 11 \cdot 5 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 121 + 25 - 2 \cdot 11 \cdot 5 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 68.22\)
\(c = \sqrt{68.22} = 8.26\)
\(c^2 = 121 + 25 - 2 \cdot 11 \cdot 5 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 68.22\)
\(c = \sqrt{68.22} = 8.26\)
Answer: c = 8.26
Question 36
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 7
• side b = 7
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 7
• side b = 7
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 7^2 + 7^2 - 2 \cdot 7 \cdot 7 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 49 + 49 - 2 \cdot 7 \cdot 7 \cdot \frac{1}{2}\)
\(c^2 = 49\)
\(c = \sqrt{49} = 7\)
\(c^2 = 49 + 49 - 2 \cdot 7 \cdot 7 \cdot \frac{1}{2}\)
\(c^2 = 49\)
\(c = \sqrt{49} = 7\)
Answer: c = 7
Question 37
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 11
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 8
• side b = 11
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 8^2 + 11^2 - 2 \cdot 8 \cdot 11 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 64 + 121 - 2 \cdot 8 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 97\)
\(c = \sqrt{97} = 9.85\)
\(c^2 = 64 + 121 - 2 \cdot 8 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 97\)
\(c = \sqrt{97} = 9.85\)
Answer: c = 9.85
Question 38
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 10
• side b = 7
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 10
• side b = 7
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 10^2 + 7^2 - 2 \cdot 10 \cdot 7 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 100 + 49 - 2 \cdot 10 \cdot 7 \cdot \frac{1}{2}\)
\(c^2 = 79\)
\(c = \sqrt{79} = 8.89\)
\(c^2 = 100 + 49 - 2 \cdot 10 \cdot 7 \cdot \frac{1}{2}\)
\(c^2 = 79\)
\(c = \sqrt{79} = 8.89\)
Answer: c = 8.89
Question 39
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 9
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 8
• side b = 9
• Angle C (between the sides) = 45°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 8^2 + 9^2 - 2 \cdot 8 \cdot 9 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 64 + 81 - 2 \cdot 8 \cdot 9 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 43.18\)
\(c = \sqrt{43.18} = 6.57\)
\(c^2 = 64 + 81 - 2 \cdot 8 \cdot 9 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 43.18\)
\(c = \sqrt{43.18} = 6.57\)
Answer: c = 6.57
Question 40
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 10
• side b = 5
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 10
• side b = 5
• Angle C (between the sides) = 60°
Find side c.
Explanation:
Solution - Law of Cosines in triangle ABC:
📝 Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
🔢 Step 1: Substitute the data
🔢 Step 2: Calculate
\(c^2 = 10^2 + 5^2 - 2 \cdot 10 \cdot 5 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 100 + 25 - 2 \cdot 10 \cdot 5 \cdot \frac{1}{2}\)
\(c^2 = 75\)
\(c = \sqrt{75} = 8.66\)
\(c^2 = 100 + 25 - 2 \cdot 10 \cdot 5 \cdot \frac{1}{2}\)
\(c^2 = 75\)
\(c = \sqrt{75} = 8.66\)
Answer: c = 8.66