To solve the questions, we will use the Law of Sines for both triangles.
For Question 2.2:
Given:
- Angle A = 39°
- Angle C = 46°
- Side BC = 2 cm
- Side AB = x cm
First, we find angle B: \[ \text{Angle B} = 180° - \text{Angle A} - \text{Angle C} = 180° - 39° - 46° = 95° \]
Now we apply the Law of Sines: \[ \frac{AB}{\sin C} = \frac{BC}{\sin A} \] This can be set up as: \[ \frac{x}{\sin(46°)} = \frac{2}{\sin(39°)} \]
Now, we can solve for x: \[ x = \frac{2 \cdot \sin(46°)}{\sin(39°)} \]
Calculating the values: \[ \sin(46°) \approx 0.7193 \] \[ \sin(39°) \approx 0.6293 \]
Now substituting these values into the equation: \[ x = \frac{2 \cdot 0.7193}{0.6293} \approx \frac{1.4386}{0.6293} \approx 2.28 \]
Rounding to one decimal place: \[ \boxed{2.3} \]
For Question 2.3:
Given:
- Angle C = 60°
- Side AC = 9 cm
- Side BC = 7 cm
First, we need to find angle A. We will again apply the Law of Sines: \[ \frac{AC}{\sin B} = \frac{BC}{\sin C} \] We need to find angle B first: Since the angles in a triangle add up to 180°: \[ \text{Angle A} + \text{Angle B} + 60° = 180° \] \[ \text{Angle A} + \text{Angle B} = 120° \] This means: \[ \text{Angle B} = 120° - \text{Angle A} \]
Now applying the Law of Sines: \[ \frac{9}{\sin(120° - A)} = \frac{7}{\sin(60°)} \]
Since \(\sin(60°) = \frac{\sqrt{3}}{2} \approx 0.8660\), we can substitute: \[ \frac{9}{\sin(120° - A)} = \frac{7}{0.8660} \]
Cross-multiply: \[ 9 \cdot 0.8660 = 7 \cdot \sin(120° - A) \] \[ 7 \cdot \sin(120° - A) \approx 7.794 \] Now solving for \(\sin(120° - A)\): \[ \sin(120° - A) \approx \frac{7.794}{7} \approx 1.114 \]
Since the sin value cannot exceed 1, we need to consider the other approach: We continue from the Law of Sines directly: Taking angle A directly. We can find angle B using side ratios since B + A = 120°.
Using trial and error or numerical methods, the Law of Cosines might be a better approach:
Using the known sides and the angle:
- Calculate using Law of Cosines: For side AB: \[ AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(60°) \] Substituting values gives: \[ AB^2 = 9^2 + 7^2 - 2 \cdot 9 \cdot 7 \cdot 0.5 \] \[ = 81 + 49 - 63 \] \[ = 67 \] Thus, \[ AB = \sqrt{67} \approx 8.2 \] Therefore, round off: \[ \boxed{8.2} \]