We have here a right triangle. We can use trigonometric ratios (sine, cosine, or tangent) to find the length of any side of the triangle.
To do this, we can use angle 77 as a reference angle to find the side JK (the value of x, in this case.)
It is also important the largest side of a right triangle is the hypotenuse (JK).
Having this information, we can proceed as follows:
LJ = 6.9 feet
x = ?
We know that the opposite side to the angle 77 is the side LJ, and knowing this, we can use the sine ratio:
[tex]\sin (77)=\frac{opp}{hyp}[/tex]
Sine is the opposite side over the hypotenuse, then we have:
[tex]\sin (77)=\frac{6.9ft}{x}[/tex]
Solving for x, we need to multiply the equation by x to both sides of the equation:
[tex]x\cdot\sin (77)=\frac{6.9ft}{x}\cdot x\Rightarrow x\cdot\sin (77)=6.9ft\cdot\frac{x}{x}[/tex]
Since x/x = 1. We can now divide both sides of the equation by sin(77):
[tex]x\cdot\frac{\sin(77)}{\sin(77)}=\frac{6.9ft}{\sin(77)}\Rightarrow x=\frac{6.9ft}{\sin (77)}[/tex]
And the value for x (the side JK) is:
[tex]x=\frac{6.9ft}{0.974370064785}\Rightarrow x=7.08149834377ft[/tex]
Rounding to the nearest tenth, we have that the value for x, the length of JK = 7.1 feet.
