Answer:
1. Triangle: B = 47.0° , C = 103.05° , c = 2.53 cm
2. Lake: c = 1105.31 ft
Step-by-step explanation:
Law of Sine Formula:[tex]\frac{sin(A)}{A} = \frac{sin(B)}{B} = \frac{sin(C)}{C}[/tex]
Given: A = 30° , a = 1.3 cm , b = 1.9 cm
[tex]\frac{sin(30)}{1.3} = \frac{sin(B)}{1.9} = \frac{sin(C)}{C}[/tex]
Solving for sin(B). Cross Multiply.
[tex]\frac{sin(30)}{1.3} = \frac{sin(B)}{1.9}\\1.3*sin(B)=1.9*sin(30)\\sin(B)=\frac{1.9*sin(30)}{1.3} \\[/tex]
B = sin^-1( [tex]\frac{1.9*sin(30)}{1.3}[/tex] )
B ≈ 46.9509202
B = 47.0°
Solve for C°
A° + B° + C° = 180°
30° + 46.95° + C° = 180°
C° = 180° - 30° - 46.95°
C° = 103.05°
Solve for sin(C)
[tex]\frac{sin(30)}{1.3} = \frac{sin(103.05)}{C}\\[/tex]
Cross Multiply
[tex]C*sin(30)=1.3*sin(103.05)\\C=\frac{1.3*sin(103.05)}{sin(30)}[/tex]
C ≈ 2.532850806
C = 2.53 cm
Law of Cosine Formula: [tex]c^2=a^2+b^2-2*a*b*cos(C)[/tex]
Given: a = 850 ft , b = 960 ft , C=75°
Solve for c.
[tex]c^2=a^2+b^2-2*a*b*cos(C)\\c^2=(850ft)^2+(960ft)^2-2*(850ft)*(960ft)*cos(75)\\\\c=\sqrt{(850ft)^2+(960ft)^2-2*(850ft)*(960ft)*cos(75)\\} \\[/tex]
c ≈ 1105.308698
c = 1105.31 ft
