Respuesta :

mhanifa

Answer:

  • a = 16.52 cm
  • b = 23.85 cm

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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)
semsee45

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:

[tex]\sf \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]

Substitute the given values into the Sine Rule formula:

[tex]\implies \sf \dfrac{a}{\sin 42^{\circ}}=\dfrac{b}{\sin 75^{\circ}}=\dfrac{22}{\sin 63^{\circ}}[/tex]

Finding side a:

[tex]\implies \sf \dfrac{a}{\sin 42^{\circ}}=\dfrac{22}{\sin 63^{\circ}}[/tex]

[tex]\implies \sf a=\dfrac{22\sin 42^{\circ}}{\sin 63^{\circ}}[/tex]

[tex]\implies \sf a=16.52162239...[/tex]

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

Finding side b:

[tex]\implies \sf \dfrac{b}{\sin 75^{\circ}}=\dfrac{22}{\sin 63^{\circ}}[/tex]

[tex]\implies \sf b=\dfrac{22\sin 75^{\circ}}{\sin 63^{\circ}}[/tex]

[tex]\implies \sf b=23.84984577...[/tex]

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

Learn more about the Sine Rule here:

https://brainly.com/question/28033514

https://brainly.com/question/27704834

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