Question

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1. calculate the length of side c using the sine law. show your work. 3 marks 2. calculate the measure of angle c using the sine law. show your work. 3 marks 3. solve the following triangle. show your work. 5 marks

Explanation:

Problem 1: Calculate the length of side \( c \) using the Sine Law.

Step 1: Find the third angle

The sum of angles in a triangle is \( 180^\circ \). Given angles \( 70^\circ \) and \( 60^\circ \), the third angle \( A = 180^\circ - 70^\circ - 60^\circ = 50^\circ \).

Step 2: Apply the Sine Law

The Sine Law states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Let \( a = 16 \) cm (opposite \( 50^\circ \)), \( C = 60^\circ \), so \( \frac{16}{\sin 50^\circ} = \frac{c}{\sin 60^\circ} \). Solve for \( c \): \( c = \frac{16 \cdot \sin 60^\circ}{\sin 50^\circ} \approx \frac{16 \cdot 0.8660}{0.7660} \approx 18.0 \) cm.

Step 1: Apply the Sine Law

Sine Law: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Let \( a = 54 \) cm (opposite \( A \)), \( b = 32 \) cm (opposite \( B = 40^\circ \)), so \( \frac{54}{\sin C} = \frac{32}{\sin 40^\circ} \).

Step 2: Solve for \( \sin C \)

\( \sin C = \frac{54 \cdot \sin 40^\circ}{32} \approx \frac{54 \cdot 0.6428}{32} \approx 1.085 \). Wait, \( \sin C \) cannot exceed 1. This means no solution or an error (maybe triangle is invalid, or mislabeled). Assuming typo, if \( a = 32 \), \( b = 54 \), then \( \sin C = \frac{32 \cdot \sin 40^\circ}{54} \approx 0.383 \), so \( C \approx 22.5^\circ \) or \( 157.5^\circ \) (but \( 157.5 + 40 > 180 \), so \( 22.5^\circ \)).

Step 1: Apply Sine Law for angle \( Z \)

Sine Law: \( \frac{XZ}{\sin Y} = \frac{XY}{\sin Z} \). \( XZ = 35 \) cm, \( XY = 24 \) cm, \( Y = 65^\circ \). So \( \frac{35}{\sin 65^\circ} = \frac{24}{\sin Z} \). \( \sin Z = \frac{24 \cdot \sin 65^\circ}{35} \approx \frac{24 \cdot 0.9063}{35} \approx 0.629 \), so \( Z \approx 39.0^\circ \) or \( 141.0^\circ \) (but \( 141 + 65 > 180 \), so \( 39.0^\circ \)).

Step 2: Find angle \( X \)

\( X = 180^\circ - 65^\circ - 39^\circ = 76^\circ \).

Step 3: Find side \( ZY \) (let \( z = ZY \))

Sine Law: \( \frac{z}{\sin X} = \frac{XY}{\sin Z} \). \( z = \frac{24 \cdot \sin 76^\circ}{\sin 39^\circ} \approx \frac{24 \cdot 0.9703}{0.6293} \approx 37.0 \) cm.

Answer:

\( \approx 18.0 \) cm

Problem 2: Calculate the measure of angle \( C \) using the Sine Law.