To solve these problems, we will use the Law of Sines or Law of Cosines as needed. Here's the step-by-step solution for each triangle:

a) Triangle a

Given:

• Angle 78°
• Angle 32°
• Side opposite 32°: 8 cm

Since the sum of angles in a triangle is 180°, the third angle is: $180° - 78° - 32° = 70°$

Using the Law of Sines: $\frac{a}{\sin(78°)} = \frac{8}{\sin(32°)}$ Solving for $a$: $a = \frac{8 \times \sin(78°)}{\sin(32°)}$

b) Triangle b

Given:

• Angle 105°
• Angle 40°
• Side opposite 105°: 10 cm

The third angle is: $180° - 105° - 40° = 35°$

Using the Law of Sines: $\frac{b}{\sin(40°)} = \frac{10}{\sin(105°)}$ Solving for $b$: $b = \frac{10 \times \sin(40°)}{\sin(105°)}$

c) Triangle c

Given:

• Angle 35°
• Angle 110°
• Side opposite 35°: 74 mm

The third angle is: $180° - 35° - 110° = 35°$

Using the Law of Sines: $\frac{c}{\sin(35°)} = \frac{74}{\sin(110°)}$ Solving for $c$: $c = \frac{74 \times \sin(35°)}{\sin(110°)}$

d) Triangle d

Given:

• Angle 48°
• Angle 50°
• Side opposite 50°: 6.2 cm

The third angle is: $180° - 48° - 50° = 82°$

Using the Law of Sines: $\frac{d}{\sin(48°)} = \frac{6.2}{\sin(50°)}$ Solving for $d$: $d = \frac{6.2 \times \sin(48°)}{\sin(50°)}$

Let's calculate these values now.The calculated lengths for each unknown side, correct to 3 significant figures, are:

• a) $a = 14.8$ cm
• b) $b = 6.66$ cm
• c) $c = 45.2$ mm
• d) $d = 6.02$ cm

Do you have any further questions or would you like more details?

Here are 5 related questions you might want to explore:

1. How does the Law of Sines work with different types of triangles?
2. When would you use the Law of Cosines instead of the Law of Sines?
3. How do you find the unknown angle in a triangle when given two sides?
4. How can we verify our answers in such problems?
5. What are the applications of trigonometry in real-life problems?

Tip: Always double-check the sum of the angles in any triangle problem to ensure it equals 180°.