6. In $$ riangle A B C $$, two sides and an angle are given. Determine the value of $$ \angle C $$ to the nearest degree and the length of $$ b $$ to the nearest tenth of a centimetre. a) $$ a=2.4 \mathrm{~cm}, c=3.2 \mathrm{~cm}, \angle A=28^{\circ} $$ b) $$ a=9.9 \mathrm{~cm}, c=11.2 \mathrm{~cm}, \angle A=58^{\circ} $$ c) $$ a=8.6 \mathrm{~cm}, c=9.4 \mathrm{~cm}, \angle A=47^{\circ} $$ d) $$ a=5.5 \mathrm{~cm}, c=10.4 \mathrm{~cm}, \angle A=30^{\circ} $$

Question

6. In $$ 	riangle A B C $$, two sides and an angle are given. Determine the value of $$ \angle C $$ to the nearest degree and the length of $$ b $$ to the nearest tenth of a centimetre. a) $$ a=2.4 \mathrm{~cm}, c=3.2 \mathrm{~cm}, \angle A=28^{\circ} $$ b) $$ a=9.9 \mathrm{~cm}, c=11.2 \mathrm{~cm}, \angle A=58^{\circ} $$ c) $$ a=8.6 \mathrm{~cm}, c=9.4 \mathrm{~cm}, \angle A=47^{\circ} $$ d) $$ a=5.5 \mathrm{~cm}, c=10.4 \mathrm{~cm}, \angle A=30^{\circ} $$

UpStudy AI · Accepted Answer

To solve for the angle $$ \angle C $$ and the length of side $$ b $$ in each triangle, we can use the Law of Sines and the Law of Cosines.

The Law of Sines states:
$$
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
$$

The Law of Cosines states:
$$
c^2 = a^2 + b^2 - 2ab \cos C
$$

We will use these laws to find $$ \angle C $$ and side $$ b $$ for each case.

### a) $$ a=2.4 \mathrm{~cm}, c=3.2 \mathrm{~cm}, \angle A=28^{\circ} $$

1. **Find $$ \angle C $$ using the Law of Sines:**
   $$
   \frac{a}{\sin A} = \frac{c}{\sin C}
   $$
   $$
   \frac{2.4}{\sin 28^\circ} = \frac{3.2}{\sin C}
   $$
   $$
   \sin C = \frac{3.2 \cdot \sin 28^\circ}{2.4}
   $$
   Calculate $$ \sin 28^\circ $$:
   $$
   \sin 28^\circ \approx 0.4695
   $$
   Now substitute:
   $$
   \sin C = \frac{3.2 \cdot 0.4695}{2.4} \approx 0.607
   $$
   Now find $$ C $$:
   $$
   C \approx \sin^{-1}(0.607) \approx 37.4^\circ
   $$

2. **Find $$ b $$ using the Law of Sines:**
   $$
   \frac{b}{\sin B} = \frac{2.4}{\sin 28^\circ}
   $$
   First, find $$ B $$:
   $$
   B = 180^\circ - A - C \approx 180^\circ - 28^\circ - 37.4^\circ \approx 114.6^\circ
   $$
   Now find $$ b $$:
   $$
   \frac{b}{\sin 114.6^\circ} = \frac{2.4}{\sin 28^\circ}
   $$
   Calculate $$ \sin 114.6^\circ $$:
   $$
   \sin 114.6^\circ \approx 0.9063
   $$
   Now substitute:
   $$
   b = \frac{2.4 \cdot 0.9063}{0.4695} \approx 4.6 \text{ cm}
   $$

### b) $$ a=9.9 \mathrm{~cm}, c=11.2 \mathrm{~cm}, \angle A=58^{\circ} $$

1. **Find $$ \angle C $$:**
   $$
   \frac{9.9}{\sin 58^\circ} = \frac{11.2}{\sin C}
   $$
   Calculate $$ \sin 58^\circ $$:
   $$
   \sin 58^\circ \approx 0.8480
   $$
   Now substitute:
   $$
   \sin C = \frac{11.2 \cdot 0.8480}{9.9} \approx 0.964
   $$
   Find $$ C $$:
   $$
   C \approx \sin^{-1}(0.964) \approx 74.5^\circ
   $$

2. **Find $$ b $$:**
   $$
   B = 180^\circ - 58^\circ - 74.5^\circ \approx 47.5^\circ
   $$
   Now find $$ b $$:
   $$
   \frac{b}{\sin 47.5^\circ} = \frac{9.9}{\sin 58^\circ}
   $$
   Calculate $$ \sin 47.5^\circ $$:
   $$
   \sin 47.5^\circ \approx 0.7314
   $$
   Now substitute:
   $$
   b = \frac{9.9 \cdot 0.7314}{0.8480} \approx 8.3 \text{ cm}
   $$

### c) $$ a=8.6 \mathrm{~cm}, c=9.4 \mathrm{~cm}, \angle A=47^{\circ} $$

1. **Find $$ \angle C $$:**
   $$
   \frac{8.6}{\sin 47^\circ} = \frac{9.4}{\sin C}
   $$
   Calculate $$ \sin 47^\circ $$:
   $$
   \sin 47^\circ \approx 0.7314
   $$
   Now substitute:
   $$
   \sin C = \frac{9.4 \cdot 0.7314}{8.6} \approx 0.785
   $$
   Find $$ C $$:
   $$
   C \approx \sin^{-1}(0.785) \approx 51.5^\circ
   $$

2. **Find $$ b $$:**
   $$
   B = 180^\circ - 47^\circ - 51.5^\circ \approx 81.5^\circ
   $$
   Now find $$ b $$:
   $$
   \frac{b}{\sin 81.5^\circ} = \frac{8.6}{\sin 47^\circ}
   $$
   Calculate $$ \sin 81.5^\circ $$:
   $$
   \sin 81.5^\circ \approx 0.9945
   $$
   Now substitute:
   $$
   b = \frac{8.6 \cdot 0.9945}{0.7314} \approx 11.6 \text{ cm}
   $$

### d) $$ a=5.5 \mathrm{~cm}, c=10.4 \mathrm{~cm}, \angle A=30^{\circ} $$

1. **Find $$ \angle C $$:**
   $$
   \frac{5.5}{\sin 30^\circ} = \frac{10.4}{\sin C}
   $$
   Calculate $$ \sin 30^\circ $$:
   $$
   \sin 30^\circ = 0.5
   $$
   Now substitute:
   $$
   \sin C = \frac{10.4 \cdot 0.5}{5.5} \approx 0.945
   $$
   Find $$ C $$:
   $$
   C \approx \sin^{-1}(0.945) \approx 71.5^\circ
   $$

2. **Find $$ b $$:**
   $$
   B = 180^\circ - 30^\circ - 71.5^\circ \approx 78.5^\circ
   $$
   Now find $$ b $$:
   $$
   \frac{b}{\sin 78.5^\circ} = \frac{5.5}{\sin 30^\circ}
   $$
   Calculate $$ \sin 78.5^\circ $$:
   $$
   \sin 78.5^\circ \approx 0.9781
   $$
   Now substitute:
   $$
   b = \frac{5.5 \cdot 0.9781}{0.5} \approx 10.7 \text{ cm}
   $$

### Summary of Results:
- a) $$ \angle C \approx 37^\circ $$, $$ b \approx 4.6 \text{ cm} $$
- b) $$ \angle C \approx 75^\circ $$, $$ b \approx 8.3 \text{ cm} $$
- c) $$ \angle C \approx 52^\circ $$, $$ b \approx 11.6 \text{ cm} $$
- d) $$ \angle C \approx 72^\circ $$, $$ b \approx 10.7 \text{ cm} $$
a) $$ \angle C \approx 37^\circ $$, $$ b \approx 4.6 \text{ cm} $$ b) $$ \angle C \approx 75^\circ $$, $$ b \approx 8.3 \text{ cm} $$ c) $$ \angle C \approx 52^\circ $$, $$ b \approx 11.6 \text{ cm} $$ d) $$ \angle C \approx 72^\circ $$, $$ b \approx 10.7 \text{ cm} $$

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