Use the result from Exercise 80 to find the acute angle betwee...

Question

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

5x - 2y + 4 = 0, 3x + 5y = 6

Accepted Answer

Hello, everyone. We are asked to determine the acute angle between the pair of lines given below. We will round our answer to the nearest 10th of a degree. We are given the equation seven X minus three, Y plus eight equals zero. And the equation six X plus Y equals 11. Our answer choices are a 56.7 degrees B 32.7 degrees C 23.6 degrees D 67.3 degrees. First, I recall before I can try to find the angle, I need to find the slope of these lines. So our first line is seven X minus three Y plus eight equals zero. And we want to find the slope of this. I'm gonna rearrange this to be in standard form. So I'm gonna subtract eight from both sides. And this makes my equation seven X minus three Y equals negative eight. And it's now in standard form where we have A plus B equals C I recall in this form, slope which is denoted by M is the opposite of A divided by B. So in our example, A is seven. So the opposite of seven is negative seven divided by B which notice in our format is a positive. And since it's a minus, we're going to treat it as negative three. This makes our slope a positive seven thirds. And I'm gonna call the M sub one just because we will have two different slopes. For our second equation, we have six X plus Y equals 11. Again, this is now in standard form. So A XBY equals C. So our slope is the opposite of a. So the opposite of six, which is negative six divided by BB is the value in front of Y. Here, we don't see one. So there's an implied one as the coefficient. So this means our slope here is negative six. And I'm gonna call that M sub two just because we have two slopes from there. I recall that we have a formula that says the tangent of the equals the absolute value of slope M sub two minus M sub one divided by one plus M sub one multiplied by M sub two. So we can use this to find our angle. So let's plug in the values we have the tangent of theta will equal the absolute value of M sub two which is negative six minus M sub one, which is seven thirds divided by one plus multiplying together seven thirds and negative six. So let's do out the math, we can, I'm going to rewrite negative six as negative 18 3rd minus seven thirds. In my numerator, in my denominator, I have one plus and now I'm gonna multiply I'm gonna treat negative six as negative 6/1. So I can cross reduce the three cancels out to a one. The negative six reduces to a negative two and seven. Multiplied by negative two is negative 14. Combining the like terms of my numerator. I will have negative 25 3rd divided by negative 13 in the denominator. Still in an absolute value symbol, I have negative 25 3rd divided by negative 13. I wrote it horizontally and now I'm gonna change that to multiply by the reciprocal. So I have the absolute value of negative 25 3rd times negative 1/13. We get the absolute value of positive 25 39th. So this means that the tangent of theta equals 25 39th, but that's not an angle. That's what the tangent of the equals. So I'm gonna use a calculator and with my calculator, I need to first make sure we are in degree mode. And then what I'm going to type in is the tangent of race to the negative first power of 25 39th. And I get 32.661. Since we want this rounded to the nearest 10th of a degree, I will round this to 32.7 degrees. And that is our acute angle and that matches answer choice B 32.7 degrees have a nice day.

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
5x - 2y + 4 = 0, 3x + 5y = 6

Key Concepts

Slope of a Line

Tangent of an Angle

Acute Angle

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