Question

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Answer 1

Answer:

a = 16.5 cm (nearest tenth)

b = 23.8 cm (nearest tenth)

Step-by-step explanation:

Definition

A, B and C are the angles and a, b and c are the sides opposite the angles.

From inspection of the triangle:

• A = 42°
• B = 75°
• c = 22 cm

Interior angles of a triangle sum to 180°:

⇒ A + B + C = 180°

⇒ 42° + 75° + C = 180°

⇒ C = 180° -  117°

⇒ C = 63°

To find the missing side lengths a and b, use the Sine Rule:

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

Substitute the given values into the Sine Rule formula:

$\implies \sf \dfrac{a}{\sin 42^{\circ}}=\dfrac{b}{\sin 75^{\circ}}=\dfrac{22}{\sin 63^{\circ}}$

Finding side a:

$\implies \sf \dfrac{a}{\sin 42^{\circ}}=\dfrac{22}{\sin 63^{\circ}}$

$\implies \sf a=\dfrac{22\sin 42^{\circ}}{\sin 63^{\circ}}$

$\implies \sf a=16.52162239...$

Therefore, side a is 16.5 cm (nearest tenth).

Finding side b:

$\implies \sf \dfrac{b}{\sin 75^{\circ}}=\dfrac{22}{\sin 63^{\circ}}$

$\implies \sf b=\dfrac{22\sin 75^{\circ}}{\sin 63^{\circ}}$

$\implies \sf b=23.84984577...$

Therefore, side b is 23.8 cm (nearest tenth).

Learn more about the Sine Rule here:

brainly.com/question/28033514

brainly.com/question/27704834

Answer 2

Answer:

• a = 16.52 cm
• b = 23.85 cm

=========================

Given

• AB = 22 cm
• m∠A = 42°
• m∠B = 75°

To find

• Side length of a and b

Solution

The law of sines is:

• a/sin A = b/sin B = c/sin C

Step 1

Find the value of ∠C using the angle sum theorem:

• m∠A + m∠B + m∠C = 180°
• 42° + 75° + C = 180°
• 117° + C = 180°
• C = 180° - 117°
• C = 63°

Step 2

Find side a:

Note, the sides of a triangle are marked as:

• a = BC, b = AC, c = AB (opposite to respective angle).

Use two pairs with one unknown:

• a/sin A = c/sin C

Substitute values and solve for a:

• a/ sin 42° = 22/sin 63°
• a = 22*sin 42°/sin 63°
• a = 16.52 cm (rounded)

Step 3

Find side b, similar to finding side a:

• b/ sin 75° = 22/sin 63°
• b = 22*sin 75°/sin 63°
• b = 23.85 cm (rounded)