Find the points on the curve y = tan x, -π/2

Question

Find the points on the curve y = tan x, -π/2 < x < π/2, where the normal line is parallel to the line y = -x/2. Sketch the curve and normal lines together, labeling each with its equation.

Accepted Answer

Hello. In this video, we are going to be determining the X coordinate where the curve Y is equal to sin of X has a normal line that is parallel to the line Y is equal to 2X minus 5. We are then going to draw the graph of the curve and the normal line. So, in this problem, we want to find a normal line that is parallel to the line Y is equal to 2 X minus 5. Now, in order to find the normal line, the first thing we need to do is we need to find the tangent line of our given curve, Y is equal to sin of X. But in order to find the tangent line, we first need to find the equation of the derivative. So, let's go ahead and take the derivative of this problem. The derivative of Y equals the sin of X is going to be Y equal to cosine of X. This is going to be the equation that gives us the slope of the tangent line, but in order to get the slope of the normal line, we need to take the negative reciprocal of the slope of the tangent line. The slope of the normal line is going to be the negative reciprocal of cosine of X, which will be -1 divided by cosine of X, and this can be further rewritten as negative sequent of X. Now, since we want this normal line to be parallel to the given line, Y is equal to 2X minus 5, what this means is that we want this normal line to be the same as the slope of the given line. Y equal to 2 X minus 5 has a slope of 2. So we want negative sequent of X to equal to 2. For this equation, we are going to divide both sides by -1, allowing us to rewrite the equation as sequent of X equal to -2, and in the region where X is between 0. And i, we need to ask ourselves, what is an angle within this region that we can plug in a sequent that will give us the value of negative 2? Well, that angle is going to be X is equal to 32. Divided by 3. This is going to be the angle between 0 and pi in which sequent is going to equal to -2. Now that we have the X coordinate that will cause the neural line to be parallel to the given line, we need to find the Y coordinate. In order to do this, we are going to take the value of X and plug it back into our original equation. That is going to give us Y is equal to sine of 2 pi divided by 3, and sine of 2 pi divided by 3 will simplify to be positive square root 3 divided by 2. So we have a point that is going to be 2 divided by 3, square root 3 divided by 2. And now that we have the point where the X coordinate causes the normal line to be parallel to the given line, we will go ahead and find the equation of the line by using the point slope form. Now again we want our slope to equal. To 2, and we just calculated the point that we are interested in. So using the point slope form, we can write the equation of the graph to be Y minus the Y coordinate, which is square root 3 divided by 2, equal to the slope M, which is 2, multiplied by X minus the X coordinate, which is 2 divided by 3. We will go ahead and distribute 2 into the quantity X minus 2 pi divided by 3, and that is going to give us Y minus the square root 3 divided by 2. Equal to 2 X minus 4 pi divided by 3. And adding square root 3 divided by 2 to both sides of the equation will give us the equation of the line, Y is equal to 2 X minus 4 pi divided by 3, plus square root 3 divided by 2. This is going to be the equation of the line, where the normal line is parallel to the given line, Y is equal to 2X minus 5. Now that we have this equation, let's go ahead and plot the normal line with the given curve. Now, our original curve is Y is equal to sin of X, but we only want the values of X between 0 and pi. Sign begins at the origin at 0.0. And it is also 0 at the value of pie. One more thing to note is that sine is equal to 1 at the value pi divided by 2. So the curve, Y is equal to sin of X between 0 and pi is going to look like this. Now we just need to go ahead and graph the equation of the line. Now, for our line, we calculated that we have Y is equal to 2X minus 4 pi divided by 3, plus the square root 3 divided by 2. This equation of the line starts slightly below the -3 -3 value on the Y axis, and it is going to have the value at i of 2 pi divided by 3 plus square root 3 divided by 2. And that is going to be roughly above the value of 3. And if we connect the dots together, this is going to be the equation of the normal line we just calculated, and this will be our graph. So I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

Find the points on the curve y = tan x, -π/2 < x < π/2, where the normal line is parallel to the line y = -x/2. Sketch the curve and normal lines together, labeling each with its equation.

Key Concepts

Derivative of a Function

Normal Line to a Curve

Parallel Lines

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