Question

right triangle abc with right angle at c, hypotenuse ab = 966 m, angle at vertex a is 41°44, side bc is labeled a, side ac is labeled b.

Explanation:

To solve for the sides \(a\) and \(b\) in the right triangle \(ABC\) (right - angled at \(C\)) with hypotenuse \(AB = 966\space m\) and \(\angle A=41^{\circ}44'\):

Step 1: Convert the angle from degrees and minutes to decimal degrees

We know that \(1^{\circ}=60'\), so to convert \(44'\) to degrees, we calculate \(\frac{44}{60}\approx0.7333^{\circ}\).
Then \(\angle A = 41^{\circ}+ 0.7333^{\circ}=41.7333^{\circ}\)

Step 2: Solve for side \(a\) (opposite to \(\angle A\))

We use the sine function, where \(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}\).
For side \(a\) (opposite \(\angle A\)) and hypotenuse \(c = 966\space m\), \(\sin(A)=\frac{a}{c}\)
So \(a=c\times\sin(A)\)
Substitute \(c = 966\) and \(A = 41.7333^{\circ}\) into the formula:
\(a=966\times\sin(41.7333^{\circ})\)
We know that \(\sin(41.7333^{\circ})\approx\sin(41^{\circ}44')\approx0.666\) (using a calculator to find the sine of the angle)
\(a = 966\times0.666\approx643.36\space m\)

Step 3: Solve for side \(b\) (adjacent to \(\angle A\))

We use the cosine function, where \(\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}\)
For side \(b\) (adjacent to \(\angle A\)) and hypotenuse \(c = 966\space m\), \(\cos(A)=\frac{b}{c}\)
So \(b = c\times\cos(A)\)
Substitute \(c = 966\) and \(A=41.7333^{\circ}\) into the formula:
\(\cos(41.7333^{\circ})\approx0.746\) (using a calculator to find the cosine of the angle)
\(b=966\times0.746\approx720.64\space m\)

If we want to find angle \(B\):

Step 4: Find \(\angle B\)

In a right - triangle, \(\angle A+\angle B = 90^{\circ}\)
So \(\angle B=90^{\circ}-\angle A\)
\(\angle B = 90^{\circ}- 41^{\circ}44'=48^{\circ}16'\) (or in decimal degrees \(90 - 41.7333=48.2667^{\circ}\))

If we assume the question is to find the length of side \(a\):

Answer:

The length of side \(a\) is approximately \(\boldsymbol{643\space m}\) (or more precisely \(643.36\space m\))

If we assume the question is to find the length of side \(b\):