Answers:
1) Distance from Emma to Patty: 77.8 miles
2) Distance from Emma to Todd: 59.6 miles
Procedure:
You can trace a nice right triangle with the information given.
I have drawn it and attached in case you need a visualization help. Open the attached.
This is the solution, step-by-step:
1) The distance from Todd to Emma, 50 miles, is the oppositte-leg to the 40° angle.
2) Y, the distance from Emma to Patty, is the hypotenuse of the triangle.
3) X, the distance from Emma to Todd, is the adjacent-leg to the 40° angle.
4) With that information you can use the trigonometric ratios to find your unknowns:
4.1) sin(40°) = opposite-leg / hypotenuse = 50 / Y => Y = 50 / sin(40°)
Y = 77.8 miles
4.2) tan (40°) = opposite-leg / adjacent-leg= 50 / X => X = 50 / tan(40°)
X = 59.6 miles
