Question

for triangle abc, we are given that m∠a = 56°, m∠c = 30.6°, and bc = 8 cm. we can use the law of sines, as well as other properties of triangles, to determine all of the missing information. note that this question allows you to use trig functions that evaluate for angle measures in degrees. to remind yourself how to evaluate trig functions for angle measures in degrees in this program, click here.
a. we know two of the three interior angle measures in the triangle. we can use the fact that the three interior angle measures must sum to 180° to determine m∠b (in degrees).
m∠b =
degrees
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b. let’s next determine the length ab in cm.

ab =
cm
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c. there is only one remaining unknown measurement (the length ac in cm). use a similar process as shown in part (b) to determine this length. note that you will need m∠b you determined in part (a).
ac =
cm
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Explanation:

Step1: Find \( m\angle B \)

The sum of interior angles in a triangle is \( 180^\circ \). Given \( m\angle A = 56^\circ \) and \( m\angle C = 30.6^\circ \), we calculate \( m\angle B \) as:

$$ m\angle B = 180^\circ - 56^\circ - 30.6^\circ = 93.4^\circ $$

Step2: Find length \( AB \) using Law of Sines

The Law of Sines states \( \frac{AB}{\sin(m\angle C)} = \frac{BC}{\sin(m\angle A)} \). We know \( BC = 8 \, \text{cm} \), \( m\angle A = 56^\circ \), \( m\angle C = 30.6^\circ \). Substitute values:

$$ \frac{AB}{\sin(30.6^\circ)} = \frac{8}{\sin(56^\circ)} $$

Solve for \( AB \):

$$ AB = \frac{8 \cdot \sin(30.6^\circ)}{\sin(56^\circ)} $$

Calculate \( \sin(30.6^\circ) \approx 0.509 \), \( \sin(56^\circ) \approx 0.829 \):

$$ AB \approx \frac{8 \cdot 0.509}{0.829} \approx \frac{4.072}{0.829} \approx 4.91 \, \text{cm} $$

Step3: Find length \( AC \) using Law of Sines

Using \( \frac{AC}{\sin(m\angle B)} = \frac{BC}{\sin(m\angle A)} \). Substitute \( m\angle B = 93.4^\circ \), \( BC = 8 \, \text{cm} \), \( m\angle A = 56^\circ \):

$$ \frac{AC}{\sin(93.4^\circ)} = \frac{8}{\sin(56^\circ)} $$

Solve for \( AC \):

$$ AC = \frac{8 \cdot \sin(93.4^\circ)}{\sin(56^\circ)} $$

Calculate \( \sin(93.4^\circ) \approx 0.998 \):

$$ AC \approx \frac{8 \cdot 0.998}{0.829} \approx \frac{7.984}{0.829} \approx 9.63 \, \text{cm} $$

Answer:

s:
a. \( m\angle B = \boxed{93.4} \) degrees
b. \( AB \approx \boxed{4.91} \) cm
c. \( AC \approx \boxed{9.63} \) cm