This is a question that involves graphing although I'm not very good at it especially Trigonometric Functions >_<
a) Find the equation of the tangent to the curve y = sinx at the origin
For part (a) I've already found the equation its: y = 0
b) Using a graph or otherwise, state the number of solutions to the equation sinx = x
c) Let m (tangent) be a positive number. For what set of values does the equation sinx = mx have exactly 3 solutions satisfying -π ≤ x ≤ π
If y = sin x
then the slope is cos x
which is one at the origin or 45 degrees or
y = x is the equation of the tangent to y = sin x at the origin.
sin x is >/=-1 and </= +1
so this whole thing will only work between x = -1 and x = +1
Now in part a we decided that y = x hit the y = sin x curve at the origin. Does it hit anywhere else? No, look at your graph.
Again look at your graph.
If m is greater than one, the y = mx hits y = sin x only at the origin
For m = 0, it hits at x = -pi, 0, and pi
for x >0 and less than 1, it hits in the first quadrant, at zero and in the third quadrant
so
0<m<1
Sure! I can help you with that.
b) To determine the number of solutions to the equation sinx = x, you can use a graph or analyze the behavior of the sine and linear functions.
Graphically, you can plot the graphs of y = sinx and y = x on the same coordinate plane. The number of intersections between these two graphs will represent the number of solutions to the equation.
Alternatively, you can compare the slopes of the two functions. The slope of y = sinx is continuously changing, while the slope of y = x is always 1. Since the slopes are different, there will be some regions where the two functions do not intersect, and other regions where they do. By studying the behavior of the sine and linear functions, you can observe that the number of intersections is finite.
c) To find the set of values for which the equation sinx = mx has exactly 3 solutions satisfying -π ≤ x ≤ π, you need to consider the behavior of the sine function relative to the positive number m.
First, let's look at the case when m is less than 1. In this scenario, the graph of y = mx will intersect the graph of y = sinx at three distinct points. The solutions will be located between the intersections of the two graphs.
Next, let's consider the case when m is equal to 1. In this situation, the graphs of y = mx and y = sinx will coincide, resulting in an infinite number of solutions. The set of values satisfying -π ≤ x ≤ π will be all real numbers.
Finally, when m is greater than 1, the graph of y = mx will intersect the graph of y = sinx at only two distinct points, meaning there will not be three solutions in the specified range.
Therefore, the set of values for which the equation sinx = mx has exactly 3 solutions satisfying -π ≤ x ≤ π is when 0 < m < 1.
I hope this helps! Let me know if you have any further questions.