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

Question

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

Accepted Answer

Hello. In this video, we are going to by finding the equations of the tangent lines to the curve Y is equal to cotangent of X on the interval from pi divided by 2 to pi divided by 2. We want the tangent lines to be parallel to the line Y is equal to -4 X. Now, in order to find the equation of the tangent line, the first thing we need to do is we need to take the derivative of the original equation. The equation given to us is Y is equal to cotangent of X, and its derivative by definition will be Y equal to negative cosequent squad of X. This is going to be the equation that gives us the slopes of the tangent line. Now, the thing is, we want our tangent lines to be parallel to the given line Y is equal to -4 X. And what this means is that is that the lines must have the same slope of -4. So we want negative cos squad of X to equal to -4. And what we are going to do is we are going to simplify this equation. First, we will divide both sides of the equation by -1, leaving us with coC2 X equal to 4. Then we are going to take the square root of both sides of this equation, leaving us with co-sequence of X equal to positive and negative 2. Now, we want to find the values of cosse of X that are equal to positive and negative 2 on the interval from negative pi divided by 2 to pi divided by 2. So, one way that we can solve this is to first rewrite cosequent into terms of sine and cosine. Cos seeking of X can be rewritten as one divided by sin of X, and this will equal to positive and negative 2. Furthermore, if we multiply both sides of this equation by sin of X, we get 1 is equal to positive and negative 2. Multiplied by sin of X, and if we divide both sides of this equation by positive and negative 2, we get sin of X is equal to positive and negative 12. Now, we want to find the values of X that make this true on the interval from negative pi divided by 2 to pi divided by 2. This interval is describing the right half of the unit circle. Now, there are two values of X that satisfy this equation. This is going to be the angle, X is equal to pi divided by 6, and X is equal to negative pi divided by 6. So we have two X coordinates, which means that we are going to end up with two equations to the tangent line that are parallel to Y is equal to -4 X. Let's go ahead and start by finding the tangent line, when X is equal to pi divided by 6. Now, in order to find the tangent line, we need to find the corresponding Y value. In order to do this, we are going to plug in pi divided by 6 into the original equation Y equal to cotangent of X. This is going to give us Y is equal to cotangent pi divided by 6, which will further simplify to be negative square root of 3. This is going to be the corresponding Y value. And since we want the slope to equal to -4, by using the point slope form, we will have Y plus the square root of 3 equal to -4, multiplied by X minus pi divided by 6. In order to simplify this equation, we are going to distribute -4 into the quantity X minus pi divided by 6. This is going to leave us of Y plus the square root of 3 equal to -4 X plus 4 pi divided by 6, and this is going to further simplify to be 2 pi divided by 3. Finally, we will go ahead and subtract the square root of 3 to both sides of the equation, and we will have Y equal to -4 X plus 2 pi divided by 3 minus the square root of 3, and this is going to be the first equation or the first tangent line that is parallel to Y is equal to -4 X. Now we are going to repeat this process, but for the value, X is equal to negative pi divided by 6. Again, we will go ahead and get the corresponding Y value by plugging a negative pi divided by 6 into cotangent. Cotangent of negative pi divided by 6 gives us the value of square root of 3. And again, since we need to use the same slope, M is equal to -4, by the point slope form, this is going to give us Y minus the square root of 3, equal to -4, multiplied by the quantity, X minus minus negative pi divided by 6, which gives us pi divided by 6. We will distribute -4 into the quantity, and that will allow us to rewrite the equation to be Y minus the square root of 3, equal to -4 X minus 4 divided by 6, which simplifies to be -2 pi divided by 3. And finally, we will add the square root of 3 to both sides of the equation, leaving us with Y equal to -4 X minus 2 pi divided by 3. Plus the square root of 3, and this is going to be the second tangent line that is parallel to Y is equal to -4 X. Thus, we will have two solutions in total. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

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

Key Concepts

Derivative of a Function

Parallel Lines

Trigonometric Functions and Their Properties

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