The figure shows the geometric construction related to the problem.
We need to complete the figure with some extra variables.
The triangle with sides of 80 miles, x, and y is a right triangle. We can apply trigonometric ratios. For example, the sine of 34° is the ratio of the opposite side (y) and the hypotenuse (80), thus:
[tex]sin34\degree=\frac{y}{80}[/tex]
Solving for y:
[tex]y=80sin34\degree[/tex]
We'll leave the calculations for later. Now apply the cosine ratio:
[tex]cos34\degree=\frac{x}{80}[/tex]
Or, equivalently:
[tex]x=80cos34\degree[/tex]
Now focus on the upper triangle (another right triangle) with legs of length x + 66 and y. We can apply the tangent of the unknown angle to find its measure as follows:
[tex]tan\theta=\frac{x+66}{y}[/tex]
Recall the tangent ratio is the ratio between the opposite leg and the adjacent leg.
Substituting the determined values of x and y:
[tex]tan\theta=\frac{80cos34\degree+66}{80sin34\degree}[/tex]
Calculating (we need a scientific calculator):
[tex]\begin{gathered} tan\theta=\frac{66.3230+66}{44.7354} \\ \\ tan\theta=2.9579 \end{gathered}[/tex]
Now calculate the value of the angle with the inverse tangent function:
[tex]\begin{gathered} \theta=arctan(2.9579) \\ \\ \theta=71.3\degree \end{gathered}[/tex]
Answer: 71.3°
