Questions: Solve each triangle using the labels as shown in Illustration 1 sides to three significant digits and angles to the nearest tent Illustration 1 12. B=17.5°, a=345 m, c=405 m

Solve each triangle using the labels as shown in Illustration 1 sides to three significant digits and angles to the nearest tent

Illustration 1
12. B=17.5°, a=345 m, c=405 m

Transcript text: Solve each triangle using the labels as shown in Illustration 1 sides to three significant digits and angles to the nearest tent Illustration 1 12. $B=17.5^{\circ}, a=345 \mathrm{~m}, c=405 \mathrm{~m}$

Solution

Solution Steps

Step 1: Find angle A using the Law of Sines

We are given B = 17.5°, a = 345 m, and c = 405 m. We can use the Law of Sines to find angle A:

sin(A) / a = sin(B) / b = sin(C) / c

sin(A) / 345 = sin(17.5°) / b = sin(C) / 405

sin(A) = (345 * sin(17.5°)) / 405

sin(A) ≈ 0.2550

A ≈ arcsin(0.2550)

A ≈ 14.8° or A ≈ 180° - 14.8° = 165.2°

Since the sum of angles in a triangle must be 180°, and B = 17.5°, if A = 165.2°, then A + B = 182.7° > 180°, which is impossible. Therefore, A ≈ 14.8°.

Step 2: Find angle C

The sum of angles in a triangle is 180°. We know A ≈ 14.8° and B = 17.5°, so:

C = 180° - A - B

C ≈ 180° - 14.8° - 17.5°

C ≈ 147.7°

Step 3: Find side b using the Law of Sines

We can use the Law of Sines again to find side b:

b / sin(B) = c / sin(C)

b = (c * sin(B)) / sin(C)

b ≈ (405 * sin(17.5°)) / sin(147.7°)

b ≈ 226 m