Question

solve the right triangle.
b
966 m
a
c
43° 37
a
b
b=□°□ (round to the nearest integer as needed.)
a=□ m (round to the nearest integer as needed.)
b=□ m (round to the nearest integer as needed.)

Explanation:

Step1: Find angle B

In a right triangle, the sum of angles is \(180^\circ\), and \(\angle C = 90^\circ\), \(\angle A = 43^\circ37'\). So \(\angle B=90^\circ - 43^\circ37'\). Convert \(90^\circ\) to \(89^\circ60'\), then \(89^\circ60' - 43^\circ37' = 46^\circ23'\) (since \(60 - 37 = 23\) and \(89 - 43 = 46\)).

Step2: Find side a (opposite to angle A)

Using sine function: \(\sin A=\frac{a}{c}\), where \(c = 966\) m, \(A = 43^\circ37'\). First, convert \(43^\circ37'\) to decimal degrees: \(37'\div60\approx0.6167^\circ\), so \(A\approx43.6167^\circ\). Then \(a = c\times\sin A=966\times\sin(43.6167^\circ)\). Calculate \(\sin(43.6167^\circ)\approx0.690\), so \(a\approx966\times0.690\approx666\) (rounded to nearest integer).

Step3: Find side b (adjacent to angle A)

Using cosine function: \(\cos A=\frac{b}{c}\), so \(b = c\times\cos A=966\times\cos(43.6167^\circ)\). Calculate \(\cos(43.6167^\circ)\approx0.723\), so \(b\approx966\times0.723\approx699\) (rounded to nearest integer).

Answer:

\(B = 46^\circ23'\)
\(a = 666\) m
\(b = 699\) m