The slope at any point is
y' = -7cosx
The line through (0,0) and (x,-7sinx) has slope -7sinx/x. That means we need
-7sinx/x = -7cosx
x = tanx
Solutions are
x = 4.493409, 7.725252, 10.904122, ...
The slopes of the lines are
1.521, -0.899, 0.639
See the first couple of lines at
http://www.wolframalpha.com/input/?i=plot+y+%3D+-7sin%28x%29%2C+y%3D1.521x%2C+y%3D-0.899x%2C+from+0+to+15
Of the infinitely many lines that are tangent to the curve
y = −7 sin x
and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places.
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