Final answer:
To solve triangle ABC with given angles and sides, one can use the law of sines to find the missing angles and sides. After finding one of the angles using the total sum of angles in a triangle, the other can be found by the sine ratio, and then the remaining side's length can be calculated using this ratio with the known sides and angles.
Step-by-step explanation:
To solve ∆ABC, given that ∠A = 58°, b = 12 cm, c = 18 cm, we will use the law of sines and the fact that the sum of the angles in a triangle equals 180°.
Calculate the value of ∠B
Starting with the sum of the angles in a triangle, ∠A + ∠B + ∠C = 180°. Since ∠A = 58°, we can set up an equation to find ∠B, knowing that ∠C can be found using the law of sines.
Calculate the value of ∠C
Using the law of sines, ∠C can be calculated by the ratio of the known sides to the sine of their respective opposite angles. First, we need to calculate ∠B using the law of sines:
rac{sin(∠B)}{b} = rac{sin(∠A)}{c}.
Calculate the length of side a
Once we have found ∠C, we can use the law of sines again to calculate the length of side a:
rac{a}{sin(∠A)} = rac{c}{sin(∠C)} or rac{a}{sin(∠A)} = rac{b}{sin(∠B)}.
Draw the diagram for ∆ABC
The diagram would be a triangle with angles labeled as A, B, and C and sides opposite them labeled as a, b, and c, respectively. Side b would measure 12 cm, and side c would measure 18 cm. The angle at A would be marked as 58°.