The point (7, minus−5) is on the terminal side of an angle thetaθ in standard position. What is the value of  minus−2 sec thetaθ + 7 sin thetaθ rounded to 3 decimal places?

To solve this problem, we first need to find the values of sec θ and sin θ.

1. The point (7, -5) is in the fourth quadrant of the Cartesian plane. We can use these coordinates to find the radius r using the Pythagorean theorem: r = sqrt(x² + y²) = sqrt(7² + (-5)²) = sqrt(49 + 25) = sqrt(74).

2. Now, we can find the values of cos θ and sin θ. In the unit circle, cos θ = x/r and sin θ = y/r. So, cos θ = 7/sqrt(74) and sin θ = -5/sqrt(74).

3. The secant of an angle is the reciprocal of the cosine of the angle. So, sec θ = sqrt(74)/7.

4. Now, we can substitute these values into the given expression: -2 sec θ + 7 sin θ = -2(sqrt(74)/7) + 7(-5/sqrt(74)).

5. Simplifying this expression gives us a value of approximately -1.598 when rounded to three decimal places.