Law of Cosines Worksheet | Algebra II PDF Worksheets - Free Printable

Let's solve each problem using the Law of Cosines, which is:

$$
c^2 = a^2 + b^2 - 2ab\cos(C)
$$

Where:
- $a$, $b$, and $c$ are side lengths,
- $C$ is the angle opposite side $c$.

We'll use this formula to find missing sides or angles. We’ll give answers to 3 significant figures as instructed.

---

Section A

#### 1) Triangle ABC
Given:
- $AB = 7$ cm
- $AC = 9$ cm
- $\angle A = 35^\circ$
Find: $a = BC$

Using Law of Cosines:
$$
a^2 = 7^2 + 9^2 - 2(7)(9)\cos(35^\circ)
$$
$$
a^2 = 49 + 81 - 126 \cdot \cos(35^\circ)
$$
$$
\cos(35^\circ) \approx 0.8192
$$
$$
a^2 = 130 - 126 \cdot 0.8192 = 130 - 103.0192 = 26.9808
$$
$$
a = \sqrt{26.9808} \approx 5.19 \text{ cm}
$$

✔ Answer: $a = 5.19$ cm

---

#### 2) Triangle ABC
Given:
- $AB = 5$ cm
- $AC = 12$ cm
- $\angle A = 77^\circ$
Find: $a = BC$

$$
a^2 = 5^2 + 12^2 - 2(5)(12)\cos(77^\circ)
$$
$$
a^2 = 25 + 144 - 120 \cdot \cos(77^\circ)
$$
$$
\cos(77^\circ) \approx 0.2250
$$
$$
a^2 = 169 - 120 \cdot 0.2250 = 169 - 27 = 142
$$
$$
a = \sqrt{142} \approx 11.9 \text{ cm}
$$

✔ Answer: $a = 11.9$ cm

---

#### 3) Triangle ABC
Given:
- $AB = 4$ cm
- $AC = 6$ cm
- $\angle A = 61^\circ$
Find: $a = BC$

$$
a^2 = 4^2 + 6^2 - 2(4)(6)\cos(61^\circ)
$$
$$
a^2 = 16 + 36 - 48 \cdot \cos(61^\circ)
$$
$$
\cos(61^\circ) \approx 0.4848
$$
$$
a^2 = 52 - 48 \cdot 0.4848 = 52 - 23.2704 = 28.7296
$$
$$
a = \sqrt{28.7296} \approx 5.36 \text{ cm}
$$

✔ Answer: $a = 5.36$ cm

---

Section B

#### 1) Triangle XYZ
Given:
- $YZ = 10$ cm
- $XZ = 14$ cm
- $XY = 8$ cm
Find: $\angle X = x$

Use Law of Cosines:
$$
\cos(x) = \frac{YZ^2 + XZ^2 - XY^2}{2 \cdot YZ \cdot XZ}
$$
$$
\cos(x) = \frac{10^2 + 14^2 - 8^2}{2 \cdot 10 \cdot 14} = \frac{100 + 196 - 64}{280} = \frac{232}{280} = 0.8286
$$
$$
x = \cos^{-1}(0.8286) \approx 34.0^\circ
$$

✔ Answer: $x = 34.0^\circ$

---

#### 2) Triangle XYZ
Given:
- $XZ = 21$ cm
- $ZY = 15$ cm
- $XY = 29$ cm
Find: $\angle X = x$

$$
\cos(x) = \frac{XZ^2 + XY^2 - ZY^2}{2 \cdot XZ \cdot XY}
$$
Wait! Let’s label correctly.

Actually:
- Side opposite $x$ is $ZY = 15$
- Adjacent sides: $XZ = 21$, $XY = 29$

So:
$$
\cos(x) = \frac{21^2 + 29^2 - 15^2}{2 \cdot 21 \cdot 29}
= \frac{441 + 841 - 225}{1218}
= \frac{1057}{1218} \approx 0.8683
$$
$$
x = \cos^{-1}(0.8683) \approx 29.4^\circ
$$

✔ Answer: $x = 29.4^\circ$

---

#### 3) Triangle XYZ
Given:
- $XZ = 9.1$ cm
- $XY = 14.08$ cm
- $ZY = 11.4$ cm
Find: $\angle X = x$

$$
\cos(x) = \frac{XZ^2 + XY^2 - ZY^2}{2 \cdot XZ \cdot XY}
= \frac{9.1^2 + 14.08^2 - 11.4^2}{2 \cdot 9.1 \cdot 14.08}
$$
Calculate:
- $9.1^2 = 82.81$
- $14.08^2 = 198.2464$
- $11.4^2 = 129.96$
- Numerator: $82.81 + 198.2464 - 129.96 = 151.0964$
- Denominator: $2 \cdot 9.1 \cdot 14.08 = 255.776$
$$
\cos(x) = \frac{151.0964}{255.776} \approx 0.5871
$$
$$
x = \cos^{-1}(0.5871) \approx 54.0^\circ
$$

✔ Answer: $x = 54.0^\circ$

---

Section C

#### 1) Triangle PQR
Given:
- $PQ = 16$ cm
- $PR = 11$ cm
- $\angle P = 28^\circ$
Find: $x = QR$

$$
x^2 = 16^2 + 11^2 - 2(16)(11)\cos(28^\circ)
$$
$$
x^2 = 256 + 121 - 352 \cdot \cos(28^\circ)
$$
$$
\cos(28^\circ) \approx 0.8829
$$
$$
x^2 = 377 - 352 \cdot 0.8829 = 377 - 310.5608 = 66.4392
$$
$$
x = \sqrt{66.4392} \approx 8.15 \text{ cm}
$$

✔ Answer: $x = 8.15$ cm

---

#### 2) Triangle DEF
Given:
- $DE = 15$ cm
- $EF = 17$ cm
- $DF = 25$ cm
Find: $\angle D = x$

$$
\cos(x) = \frac{DE^2 + DF^2 - EF^2}{2 \cdot DE \cdot DF}
= \frac{15^2 + 25^2 - 17^2}{2 \cdot 15 \cdot 25}
= \frac{225 + 625 - 289}{750}
= \frac{561}{750} = 0.748
$$
$$
x = \cos^{-1}(0.748) \approx 41.1^\circ
$$

✔ Answer: $x = 41.1^\circ$

---

#### 3) Triangle JKL
Given:
- $JK = 7.2$ cm
- $KL = 4.9$ cm
- $\angle K = 54^\circ$
Find: $x = JL$

$$
x^2 = 7.2^2 + 4.9^2 - 2(7.2)(4.9)\cos(54^\circ)
$$
$$
x^2 = 51.84 + 24.01 - 70.56 \cdot \cos(54^\circ)
$$
$$
\cos(54^\circ) \approx 0.5878
$$
$$
x^2 = 75.85 - 70.56 \cdot 0.5878 = 75.85 - 41.43 = 34.42
$$
$$
x = \sqrt{34.42} \approx 5.87 \text{ cm}
$$

✔ Answer: $x = 5.87$ cm

---

✔ Final Answers Summary

#### Section A
1) $a = 5.19$ cm
2) $a = 11.9$ cm
3) $a = 5.36$ cm

#### Section B
1) $x = 34.0^\circ$
2) $x = 29.4^\circ$
3) $x = 54.0^\circ$

#### Section C
1) $x = 8.15$ cm
2) $x = 41.1^\circ$
3) $x = 5.87$ cm

---

All answers are rounded to 3 significant figures as requested.

Let me know if you'd like a step-by-step diagram explanation for any specific triangle!