Cosine Rule — Step 1 — Dynamic Practice
Cosine Rule — Step 1 — Dynamic Practice. Practice questions to deepen understanding of using the cosine rule to find a missing side in a triangle. Online math practice with full solutions and clear explanations.
Dynamic practice in the cosine rule (step 1) — using c² = a² + b² − 2ab·cos C to find a missing side. New questions every attempt.
Question 1
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) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 7
• 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
\(c^2 = 7^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 49 + 100 - 2 \cdot 7 \cdot 10 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 27.76\)
\(c = \sqrt{27.76} = 5.27\)
\(c^2 = 49 + 100 - 2 \cdot 7 \cdot 10 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 27.76\)
\(c = \sqrt{27.76} = 5.27\)
Answer: c = 5.27
Question 2
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) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 11
• 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
\(c^2 = 11^2 + 4^2 - 2 \cdot 11 \cdot 4 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 121 + 16 - 2 \cdot 11 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 93\)
\(c = \sqrt{93} = 9.64\)
\(c^2 = 121 + 16 - 2 \cdot 11 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 93\)
\(c = \sqrt{93} = 9.64\)
Answer: c = 9.64
Question 3
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) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 11
• 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
\(c^2 = 11^2 + 10^2 - 2 \cdot 11 \cdot 10 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 121 + 100 - 2 \cdot 11 \cdot 10 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 30.47\)
\(c = \sqrt{30.47} = 5.52\)
\(c^2 = 121 + 100 - 2 \cdot 11 \cdot 10 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 30.47\)
\(c = \sqrt{30.47} = 5.52\)
Answer: c = 5.52
Question 4
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) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 7
• 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
\(c^2 = 7^2 + 11^2 - 2 \cdot 7 \cdot 11 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 49 + 121 - 2 \cdot 7 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 93\)
\(c = \sqrt{93} = 9.64\)
\(c^2 = 49 + 121 - 2 \cdot 7 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 93\)
\(c = \sqrt{93} = 9.64\)
Answer: c = 9.64
Question 5
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) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 4
• 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
\(c^2 = 4^2 + 10^2 - 2 \cdot 4 \cdot 10 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 16 + 100 - 2 \cdot 4 \cdot 10 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 59.43\)
\(c = \sqrt{59.43} = 7.71\)
\(c^2 = 16 + 100 - 2 \cdot 4 \cdot 10 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 59.43\)
\(c = \sqrt{59.43} = 7.71\)
Answer: c = 7.71
Question 6
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 4
• side b = 7
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 4
• 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
\(c^2 = 4^2 + 7^2 - 2 \cdot 4 \cdot 7 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 16 + 49 - 2 \cdot 4 \cdot 7 \cdot \frac{1}{2}\)
\(c^2 = 37\)
\(c = \sqrt{37} = 6.08\)
\(c^2 = 16 + 49 - 2 \cdot 4 \cdot 7 \cdot \frac{1}{2}\)
\(c^2 = 37\)
\(c = \sqrt{37} = 6.08\)
Answer: c = 6.08
Question 7
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 6
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 8
• 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
\(c^2 = 8^2 + 6^2 - 2 \cdot 8 \cdot 6 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 64 + 36 - 2 \cdot 8 \cdot 6 \cdot \frac{1}{2}\)
\(c^2 = 52\)
\(c = \sqrt{52} = 7.21\)
\(c^2 = 64 + 36 - 2 \cdot 8 \cdot 6 \cdot \frac{1}{2}\)
\(c^2 = 52\)
\(c = \sqrt{52} = 7.21\)
Answer: c = 7.21
Question 8
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) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 8
• 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
\(c^2 = 8^2 + 9^2 - 2 \cdot 8 \cdot 9 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 64 + 81 - 2 \cdot 8 \cdot 9 \cdot \frac{1}{2}\)
\(c^2 = 73\)
\(c = \sqrt{73} = 8.54\)
\(c^2 = 64 + 81 - 2 \cdot 8 \cdot 9 \cdot \frac{1}{2}\)
\(c^2 = 73\)
\(c = \sqrt{73} = 8.54\)
Answer: c = 8.54
Question 9
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 10
• side b = 10
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 10
• 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
\(c^2 = 10^2 + 10^2 - 2 \cdot 10 \cdot 10 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 100 + 100 - 2 \cdot 10 \cdot 10 \cdot \frac{1}{2}\)
\(c^2 = 100\)
\(c = \sqrt{100} = 10\)
\(c^2 = 100 + 100 - 2 \cdot 10 \cdot 10 \cdot \frac{1}{2}\)
\(c^2 = 100\)
\(c = \sqrt{100} = 10\)
Answer: c = 10
Question 10
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) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 6
• 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
\(c^2 = 6^2 + 11^2 - 2 \cdot 6 \cdot 11 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 36 + 121 - 2 \cdot 6 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 63.66\)
\(c = \sqrt{63.66} = 7.98\)
\(c^2 = 36 + 121 - 2 \cdot 6 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 63.66\)
\(c = \sqrt{63.66} = 7.98\)
Answer: c = 7.98
Question 11
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 4
• side b = 4
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 4
• side b = 4
• 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
\(c^2 = 4^2 + 4^2 - 2 \cdot 4 \cdot 4 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 16 + 16 - 2 \cdot 4 \cdot 4 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 4.29\)
\(c = \sqrt{4.29} = 2.07\)
\(c^2 = 16 + 16 - 2 \cdot 4 \cdot 4 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 4.29\)
\(c = \sqrt{4.29} = 2.07\)
Answer: c = 2.07
Question 12
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 8
• side b = 6
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 8
• 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
\(c^2 = 8^2 + 6^2 - 2 \cdot 8 \cdot 6 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 64 + 36 - 2 \cdot 8 \cdot 6 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 32.12\)
\(c = \sqrt{32.12} = 5.67\)
\(c^2 = 64 + 36 - 2 \cdot 8 \cdot 6 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 32.12\)
\(c = \sqrt{32.12} = 5.67\)
Answer: c = 5.67
Question 13
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 4
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 5
• 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
\(c^2 = 5^2 + 4^2 - 2 \cdot 5 \cdot 4 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 25 + 16 - 2 \cdot 5 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 21\)
\(c = \sqrt{21} = 4.58\)
\(c^2 = 25 + 16 - 2 \cdot 5 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 21\)
\(c = \sqrt{21} = 4.58\)
Answer: c = 4.58
Question 14
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 10
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 5
• 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
\(c^2 = 5^2 + 10^2 - 2 \cdot 5 \cdot 10 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 25 + 100 - 2 \cdot 5 \cdot 10 \cdot \frac{1}{2}\)
\(c^2 = 75\)
\(c = \sqrt{75} = 8.66\)
\(c^2 = 25 + 100 - 2 \cdot 5 \cdot 10 \cdot \frac{1}{2}\)
\(c^2 = 75\)
\(c = \sqrt{75} = 8.66\)
Answer: c = 8.66
Question 15
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) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 5
• 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
\(c^2 = 5^2 + 8^2 - 2 \cdot 5 \cdot 8 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 25 + 64 - 2 \cdot 5 \cdot 8 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 19.72\)
\(c = \sqrt{19.72} = 4.44\)
\(c^2 = 25 + 64 - 2 \cdot 5 \cdot 8 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 19.72\)
\(c = \sqrt{19.72} = 4.44\)
Answer: c = 4.44
Question 16
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 17
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) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 11
• 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
\(c^2 = 11^2 + 11^2 - 2 \cdot 11 \cdot 11 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 121 + 121 - 2 \cdot 11 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 70.88\)
\(c = \sqrt{70.88} = 8.42\)
\(c^2 = 121 + 121 - 2 \cdot 11 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 70.88\)
\(c = \sqrt{70.88} = 8.42\)
Answer: c = 8.42
Question 18
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) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 6
• 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
\(c^2 = 6^2 + 5^2 - 2 \cdot 6 \cdot 5 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 36 + 25 - 2 \cdot 6 \cdot 5 \cdot \frac{1}{2}\)
\(c^2 = 31\)
\(c = \sqrt{31} = 5.57\)
\(c^2 = 36 + 25 - 2 \cdot 6 \cdot 5 \cdot \frac{1}{2}\)
\(c^2 = 31\)
\(c = \sqrt{31} = 5.57\)
Answer: c = 5.57
Question 19
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 9
• side b = 10
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 9
• 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
\(c^2 = 9^2 + 10^2 - 2 \cdot 9 \cdot 10 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 81 + 100 - 2 \cdot 9 \cdot 10 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 25.12\)
\(c = \sqrt{25.12} = 5.01\)
\(c^2 = 81 + 100 - 2 \cdot 9 \cdot 10 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 25.12\)
\(c = \sqrt{25.12} = 5.01\)
Answer: c = 5.01
Question 20
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 4
• side b = 9
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 4
• 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
\(c^2 = 4^2 + 9^2 - 2 \cdot 4 \cdot 9 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 16 + 81 - 2 \cdot 4 \cdot 9 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 34.65\)
\(c = \sqrt{34.65} = 5.89\)
\(c^2 = 16 + 81 - 2 \cdot 4 \cdot 9 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 34.65\)
\(c = \sqrt{34.65} = 5.89\)
Answer: c = 5.89
Question 21
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 7
• side b = 8
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 7
• 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
\(c^2 = 7^2 + 8^2 - 2 \cdot 7 \cdot 8 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 49 + 64 - 2 \cdot 7 \cdot 8 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 16.01\)
\(c = \sqrt{16.01} = 4\)
\(c^2 = 49 + 64 - 2 \cdot 7 \cdot 8 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 16.01\)
\(c = \sqrt{16.01} = 4\)
Answer: c = 4
Question 22
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 23
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 7
• side b = 8
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 7
• 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
\(c^2 = 7^2 + 8^2 - 2 \cdot 7 \cdot 8 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 49 + 64 - 2 \cdot 7 \cdot 8 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 33.8\)
\(c = \sqrt{33.8} = 5.81\)
\(c^2 = 49 + 64 - 2 \cdot 7 \cdot 8 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 33.8\)
\(c = \sqrt{33.8} = 5.81\)
Answer: c = 5.81
Question 24
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) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 11
• 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
\(c^2 = 11^2 + 9^2 - 2 \cdot 11 \cdot 9 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 121 + 81 - 2 \cdot 11 \cdot 9 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 30.53\)
\(c = \sqrt{30.53} = 5.53\)
\(c^2 = 121 + 81 - 2 \cdot 11 \cdot 9 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 30.53\)
\(c = \sqrt{30.53} = 5.53\)
Answer: c = 5.53
Question 25
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 6
• side b = 4
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 6
• 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
\(c^2 = 6^2 + 4^2 - 2 \cdot 6 \cdot 4 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 36 + 16 - 2 \cdot 6 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 28\)
\(c = \sqrt{28} = 5.29\)
\(c^2 = 36 + 16 - 2 \cdot 6 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 28\)
\(c = \sqrt{28} = 5.29\)
Answer: c = 5.29
Question 26
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 4
• side b = 11
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 4
• 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
\(c^2 = 4^2 + 11^2 - 2 \cdot 4 \cdot 11 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 16 + 121 - 2 \cdot 4 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 74.77\)
\(c = \sqrt{74.77} = 8.65\)
\(c^2 = 16 + 121 - 2 \cdot 4 \cdot 11 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 74.77\)
\(c = \sqrt{74.77} = 8.65\)
Answer: c = 8.65
Question 27
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 5
• side b = 5
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 5
• 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
\(c^2 = 5^2 + 5^2 - 2 \cdot 5 \cdot 5 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 25 + 25 - 2 \cdot 5 \cdot 5 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 6.7\)
\(c = \sqrt{6.7} = 2.59\)
\(c^2 = 25 + 25 - 2 \cdot 5 \cdot 5 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 6.7\)
\(c = \sqrt{6.7} = 2.59\)
Answer: c = 2.59
Question 28
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 6
• side b = 9
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 6
• 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
\(c^2 = 6^2 + 9^2 - 2 \cdot 6 \cdot 9 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 36 + 81 - 2 \cdot 6 \cdot 9 \cdot \frac{1}{2}\)
\(c^2 = 63\)
\(c = \sqrt{63} = 7.94\)
\(c^2 = 36 + 81 - 2 \cdot 6 \cdot 9 \cdot \frac{1}{2}\)
\(c^2 = 63\)
\(c = \sqrt{63} = 7.94\)
Answer: c = 7.94
Question 29
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) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 8
• 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
\(c^2 = 8^2 + 7^2 - 2 \cdot 8 \cdot 7 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 64 + 49 - 2 \cdot 8 \cdot 7 \cdot \frac{1}{2}\)
\(c^2 = 57\)
\(c = \sqrt{57} = 7.55\)
\(c^2 = 64 + 49 - 2 \cdot 8 \cdot 7 \cdot \frac{1}{2}\)
\(c^2 = 57\)
\(c = \sqrt{57} = 7.55\)
Answer: c = 7.55
Question 30
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 6
• side b = 7
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 6
• 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
\(c^2 = 6^2 + 7^2 - 2 \cdot 6 \cdot 7 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 36 + 49 - 2 \cdot 6 \cdot 7 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 12.25\)
\(c = \sqrt{12.25} = 3.5\)
\(c^2 = 36 + 49 - 2 \cdot 6 \cdot 7 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 12.25\)
\(c = \sqrt{12.25} = 3.5\)
Answer: c = 3.5
Question 31
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) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 5
• 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
\(c^2 = 5^2 + 11^2 - 2 \cdot 5 \cdot 11 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 25 + 121 - 2 \cdot 5 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 91\)
\(c = \sqrt{91} = 9.54\)
\(c^2 = 25 + 121 - 2 \cdot 5 \cdot 11 \cdot \frac{1}{2}\)
\(c^2 = 91\)
\(c = \sqrt{91} = 9.54\)
Answer: c = 9.54
Question 32
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) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 8
• 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
\(c^2 = 8^2 + 7^2 - 2 \cdot 8 \cdot 7 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 64 + 49 - 2 \cdot 8 \cdot 7 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 33.8\)
\(c = \sqrt{33.8} = 5.81\)
\(c^2 = 64 + 49 - 2 \cdot 8 \cdot 7 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 33.8\)
\(c = \sqrt{33.8} = 5.81\)
Answer: c = 5.81
Question 33
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) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 11
• 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
\(c^2 = 11^2 + 11^2 - 2 \cdot 11 \cdot 11 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 121 + 121 - 2 \cdot 11 \cdot 11 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 32.42\)
\(c = \sqrt{32.42} = 5.69\)
\(c^2 = 121 + 121 - 2 \cdot 11 \cdot 11 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 32.42\)
\(c = \sqrt{32.42} = 5.69\)
Answer: c = 5.69
Question 34
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 7
• side b = 5
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 7
• 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
\(c^2 = 7^2 + 5^2 - 2 \cdot 7 \cdot 5 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 49 + 25 - 2 \cdot 7 \cdot 5 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 24.5\)
\(c = \sqrt{24.5} = 4.95\)
\(c^2 = 49 + 25 - 2 \cdot 7 \cdot 5 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 24.5\)
\(c = \sqrt{24.5} = 4.95\)
Answer: c = 4.95
Question 35
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 6
• side b = 8
• Angle C (between the sides) = 45°
Find side c.
In triangle ABC the following are given:
• side a = 6
• 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
\(c^2 = 6^2 + 8^2 - 2 \cdot 6 \cdot 8 \cdot \cos(45°)\)
🔢 Step 2: Calculate
\(\cos(45°) = \frac{\sqrt{2}}{2}\)
\(c^2 = 36 + 64 - 2 \cdot 6 \cdot 8 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 32.12\)
\(c = \sqrt{32.12} = 5.67\)
\(c^2 = 36 + 64 - 2 \cdot 6 \cdot 8 \cdot \frac{\sqrt{2}}{2}\)
\(c^2 = 32.12\)
\(c = \sqrt{32.12} = 5.67\)
Answer: c = 5.67
Question 36
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) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 10
• 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
\(c^2 = 10^2 + 5^2 - 2 \cdot 10 \cdot 5 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 100 + 25 - 2 \cdot 10 \cdot 5 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 38.4\)
\(c = \sqrt{38.4} = 6.2\)
\(c^2 = 100 + 25 - 2 \cdot 10 \cdot 5 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 38.4\)
\(c = \sqrt{38.4} = 6.2\)
Answer: c = 6.2
Question 37
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 7
• side b = 4
• Angle C (between the sides) = 60°
Find side c.
In triangle ABC the following are given:
• side a = 7
• 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
\(c^2 = 7^2 + 4^2 - 2 \cdot 7 \cdot 4 \cdot \cos(60°)\)
🔢 Step 2: Calculate
\(\cos(60°) = \frac{1}{2}\)
\(c^2 = 49 + 16 - 2 \cdot 7 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 37\)
\(c = \sqrt{37} = 6.08\)
\(c^2 = 49 + 16 - 2 \cdot 7 \cdot 4 \cdot \frac{1}{2}\)
\(c^2 = 37\)
\(c = \sqrt{37} = 6.08\)
Answer: c = 6.08
Question 38
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) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 5
• 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
\(c^2 = 5^2 + 6^2 - 2 \cdot 5 \cdot 6 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 25 + 36 - 2 \cdot 5 \cdot 6 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 9.04\)
\(c = \sqrt{9.04} = 3.01\)
\(c^2 = 25 + 36 - 2 \cdot 5 \cdot 6 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 9.04\)
\(c = \sqrt{9.04} = 3.01\)
Answer: c = 3.01
Question 39
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) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 7
• 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
\(c^2 = 7^2 + 11^2 - 2 \cdot 7 \cdot 11 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 49 + 121 - 2 \cdot 7 \cdot 11 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 36.63\)
\(c = \sqrt{36.63} = 6.05\)
\(c^2 = 49 + 121 - 2 \cdot 7 \cdot 11 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 36.63\)
\(c = \sqrt{36.63} = 6.05\)
Answer: c = 6.05
Question 40
2.50 pts
📐 Law of Cosines in triangle ABC:
In triangle ABC the following are given:
• side a = 6
• side b = 4
• Angle C (between the sides) = 30°
Find side c.
In triangle ABC the following are given:
• side a = 6
• side b = 4
• 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
\(c^2 = 6^2 + 4^2 - 2 \cdot 6 \cdot 4 \cdot \cos(30°)\)
🔢 Step 2: Calculate
\(\cos(30°) = \frac{\sqrt{3}}{2}\)
\(c^2 = 36 + 16 - 2 \cdot 6 \cdot 4 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 10.43\)
\(c = \sqrt{10.43} = 3.23\)
\(c^2 = 36 + 16 - 2 \cdot 6 \cdot 4 \cdot \frac{\sqrt{3}}{2}\)
\(c^2 = 10.43\)
\(c = \sqrt{10.43} = 3.23\)
Answer: c = 3.23