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# Slope of a line secant to a curve

What is the slope of a line between two points on a curve?  This is called a "secant" line. Created by Sal Khan.

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• I'm having difficulty understanding the concept of a secant line as it pertains to a sine graph. Specifically, there is a problem in the "Slope of secant lines" exercise, where there are four questions. In each you are asked to evaluate the rates of change between secant lines for four different points.

The second question asks whether [Sin 5/2 Pi - Sin 1/2 Pi / 5/2 Pi - 1/2 Pi] is greater than, less than or equal to [Sin 2/3 Pi - Sin 1/3 Pi / 2/3 Pi - 1/3 Pi]. Evaluating the first part, I see that 5/2 Pi is the same as 1/2 Pi, and therefore the change in Y is zero. However, when evaluating the second part, it seems to me that Delta Y/Delta X is Sin 1/3 Pi / 1/3 Pi.

But the lesson tells me that the secant line is horizontal, which I do not understand. Can anyone help me with understanding secant lines on a sine graph? I did all of the trig lessons and most of the videos already, but if anyone can clarify this concept or point me in the direction of some valuable review materials, that would be legit. ( :
• You're correct that sin(5π/2) - sin(π/2) = 0, so the slope of the first line is 0 making it horizontal.

Let's look at the second slope. In the numerator it has sin(2π/3) - sin(π/3).
Can we simplify that?

Remember that sin x = sin(π - x), so
sin(2π/3) = sin(π - 2π/3) = sin(π/3).

The numerator then becomes
sin(2π/3) - sin(π/3) = sin(π/3) - sin(π/3) = 0.

So, both slopes are horizontal and thus equal.

Maybe the mistake you made was that you thought that
sin(2π/3) - sin(π-3) = sin(2π/3 - π/3)?
The two aren't equal.
• Does a secant line always only touch 2 points on the graph? (is this not doable for some periodic functions where it would hit it 3 times + for example?)
• As the term is typically used in calculus, a secant line intersects the curve in two places locally -- it may or may not intersect the curve somewhere else. So the requirement of just two intersections applies just to the small region of interest and is not a strict requirement for regions you are not concerned with at the moment.

Note: in some fields of geometry the requirement of exactly two points of intersection is much more strict than what we usually have in calculus.
• what is the difference between a chord and a secant?
• A chord is a line segment, a secant is a line.
• What are the main differences between a tangent line and a secant line in a curve?
• Why does he use x not and sub 1 in the first graph but sub 1 and sub 2 in the second graph? Is there any special situation that those are reserved for?
• No, they are just placeholders. Nothing is stopping you from writing x205, but for the sake of clarity he just uses smaller values.

X(anything) just means a non determined value of x.
• Can't we calculate slope in the same way as we do for lines?We have coordinates in the curve.
• At Sal mentions interval. Can someone explain to me what this 'interval' means? I keep on hearing this term more and more often.
• In this context, "interval" just refers to the section of points (along the x-axis) between x1 and x2. If it were an interval of time, for example, it could be the segment of time between two events. Hope this helps!
• Can't we call that secant line is a chord? Or is it both? Or one in specific?
(1 vote)
• At the high school level, the word 'chord' is usually reserved for a line segment across a circle. But in the broader mathematical community, 'chord' and 'secant line' are the same thing.