Law of Sines and Cosines Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem using the Law of Sines, which states:

$$
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
$$

This law relates the sides of a triangle to the sines of their opposite angles.

We'll go through each problem step by step.

---

1.

Given:
- Side $ b = 5 $
- Angle $ A = 53^\circ $
- Angle $ B = 40^\circ $
- Find side $ x $ (side $ a $)

Step 1: Find angle $ C $
$$
C = 180^\circ - 53^\circ - 40^\circ = 87^\circ
$$

Step 2: Use Law of Sines
$$
\frac{x}{\sin 53^\circ} = \frac{5}{\sin 40^\circ}
$$

$$
x = \frac{5 \cdot \sin 53^\circ}{\sin 40^\circ}
$$

Using calculator:
- $ \sin 53^\circ \approx 0.7986 $
- $ \sin 40^\circ \approx 0.6428 $

$$
x \approx \frac{5 \cdot 0.7986}{0.6428} \approx \frac{3.993}{0.6428} \approx 6.21
$$

✔ Answer: $ x \approx 6.21 $

---

2.

Given:
- Side $ c = 10 $
- Angle $ B = 128^\circ $
- Angle $ C = 32^\circ $
- Find side $ x $ (side $ b $)

Step 1: Find angle $ A $
$$
A = 180^\circ - 128^\circ - 32^\circ = 20^\circ
$$

Step 2: Use Law of Sines
$$
\frac{x}{\sin 128^\circ} = \frac{10}{\sin 32^\circ}
$$

$$
x = \frac{10 \cdot \sin 128^\circ}{\sin 32^\circ}
$$

- $ \sin 128^\circ = \sin(180^\circ - 52^\circ) = \sin 52^\circ \approx 0.7880 $
- $ \sin 32^\circ \approx 0.5299 $

$$
x \approx \frac{10 \cdot 0.7880}{0.5299} \approx \frac{7.88}{0.5299} \approx 14.85
$$

✔ Answer: $ x \approx 14.85 $

---

3.

Given:
- Side $ b = 9 $
- Angle $ A = 99^\circ $
- Angle $ C = 50^\circ $
- Find side $ x $ (side $ a $)

Step 1: Find angle $ B $
$$
B = 180^\circ - 99^\circ - 50^\circ = 31^\circ
$$

Step 2: Use Law of Sines
$$
\frac{x}{\sin 99^\circ} = \frac{9}{\sin 31^\circ}
$$

$$
x = \frac{9 \cdot \sin 99^\circ}{\sin 31^\circ}
$$

- $ \sin 99^\circ \approx 0.9877 $
- $ \sin 31^\circ \approx 0.5150 $

$$
x \approx \frac{9 \cdot 0.9877}{0.5150} \approx \frac{8.8893}{0.5150} \approx 17.26
$$

✔ Answer: $ x \approx 17.26 $

---

4.

Given:
- Side $ c = 6 $
- Angle $ A = 62^\circ $
- Angle $ B = 65^\circ $
- Find side $ x $ (side $ a $)

Step 1: Find angle $ C $
$$
C = 180^\circ - 62^\circ - 65^\circ = 53^\circ
$$

Step 2: Use Law of Sines
$$
\frac{x}{\sin 62^\circ} = \frac{6}{\sin 53^\circ}
$$

$$
x = \frac{6 \cdot \sin 62^\circ}{\sin 53^\circ}
$$

- $ \sin 62^\circ \approx 0.8829 $
- $ \sin 53^\circ \approx 0.7986 $

$$
x \approx \frac{6 \cdot 0.8829}{0.7986} \approx \frac{5.2974}{0.7986} \approx 6.64
$$

✔ Answer: $ x \approx 6.64 $

---

5.

Given:
- Side $ b = 15 $
- Angle $ A = 110^\circ $
- Angle $ C = 46^\circ $
- Find side $ x $ (side $ a $)

Step 1: Find angle $ B $
$$
B = 180^\circ - 110^\circ - 46^\circ = 24^\circ
$$

Step 2: Use Law of Sines
$$
\frac{x}{\sin 110^\circ} = \frac{15}{\sin 24^\circ}
$$

$$
x = \frac{15 \cdot \sin 110^\circ}{\sin 24^\circ}
$$

- $ \sin 110^\circ = \sin(180^\circ - 70^\circ) = \sin 70^\circ \approx 0.9397 $
- $ \sin 24^\circ \approx 0.4067 $

$$
x \approx \frac{15 \cdot 0.9397}{0.4067} \approx \frac{14.0955}{0.4067} \approx 34.65
$$

✔ Answer: $ x \approx 34.65 $

---

6.

Given:
- Side $ a = 7 $
- Angle $ B = 81^\circ $
- Angle $ C = 61^\circ $
- Find side $ x $ (side $ b $)

Step 1: Find angle $ A $
$$
A = 180^\circ - 81^\circ - 61^\circ = 38^\circ
$$

Step 2: Use Law of Sines
$$
\frac{7}{\sin 38^\circ} = \frac{x}{\sin 81^\circ}
$$

$$
x = \frac{7 \cdot \sin 81^\circ}{\sin 38^\circ}
$$

- $ \sin 81^\circ \approx 0.9877 $
- $ \sin 38^\circ \approx 0.6157 $

$$
x \approx \frac{7 \cdot 0.9877}{0.6157} \approx \frac{6.9139}{0.6157} \approx 11.23
$$

✔ Answer: $ x \approx 11.23 $

---

7. Find all missing sides and angles

Given:
- Side $ a = 12 $
- Angle $ A = 41^\circ $
- Angle $ C = 76^\circ $

Step 1: Find angle $ B $
$$
B = 180^\circ - 41^\circ - 76^\circ = 63^\circ
$$

Step 2: Use Law of Sines to find other sides

Find side $ b $ (opposite $ B $):
$$
\frac{12}{\sin 41^\circ} = \frac{b}{\sin 63^\circ}
\Rightarrow b = \frac{12 \cdot \sin 63^\circ}{\sin 41^\circ}
$$

- $ \sin 63^\circ \approx 0.8910 $
- $ \sin 41^\circ \approx 0.6561 $

$$
b \approx \frac{12 \cdot 0.8910}{0.6561} \approx \frac{10.692}{0.6561} \approx 16.31
$$

Find side $ c $ (opposite $ C $):
$$
\frac{12}{\sin 41^\circ} = \frac{c}{\sin 76^\circ}
\Rightarrow c = \frac{12 \cdot \sin 76^\circ}{\sin 41^\circ}
$$

- $ \sin 76^\circ \approx 0.9703 $

$$
c \approx \frac{12 \cdot 0.9703}{0.6561} \approx \frac{11.6436}{0.6561} \approx 17.75
$$

✔ Answers:
- Angle $ B = 63^\circ $
- Side $ b \approx 16.31 $
- Side $ c \approx 17.75 $

---

8. Find all missing sides and angles

Given:
- Side $ c = 16 $
- Angle $ A = 95^\circ $
- Angle $ C = 48^\circ $

Step 1: Find angle $ B $
$$
B = 180^\circ - 95^\circ - 48^\circ = 37^\circ
$$

Step 2: Use Law of Sines

Find side $ a $ (opposite $ A $):
$$
\frac{a}{\sin 95^\circ} = \frac{16}{\sin 48^\circ}
\Rightarrow a = \frac{16 \cdot \sin 95^\circ}{\sin 48^\circ}
$$

- $ \sin 95^\circ \approx 0.9962 $
- $ \sin 48^\circ \approx 0.7431 $

$$
a \approx \frac{16 \cdot 0.9962}{0.7431} \approx \frac{15.9392}{0.7431} \approx 21.44
$$

Find side $ b $ (opposite $ B $):
$$
\frac{b}{\sin 37^\circ} = \frac{16}{\sin 48^\circ}
\Rightarrow b = \frac{16 \cdot \sin 37^\circ}{\sin 48^\circ}
$$

- $ \sin 37^\circ \approx 0.5878 $

$$
b \approx \frac{16 \cdot 0.5878}{0.7431} \approx \frac{9.4048}{0.7431} \approx 12.65
$$

✔ Answers:
- Angle $ B = 37^\circ $
- Side $ a \approx 21.44 $
- Side $ b \approx 12.65 $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ x \approx 6.21 $ |
| 2 | $ x \approx 14.85 $ |
| 3 | $ x \approx 17.26 $ |
| 4 | $ x \approx 6.64 $ |
| 5 | $ x \approx 34.65 $ |
| 6 | $ x \approx 11.23 $ |
| 7 | $ B = 63^\circ, b \approx 16.31, c \approx 17.75 $ |
| 8 | $ B = 37^\circ, a \approx 21.44, b \approx 12.65 $ |

> Note: All values are rounded to two decimal places unless otherwise specified.

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