Video Transcript
Find the measure of the angle between the two straight lines whose direction cosines are three over 13 root two, nine over 11 root two, negative three over seven root two and negative 10 over 13 root two, negative eight over 13 root two, nine over eight root two. Give your answer to the nearest second.
In this question, we’re asked to find the measure of the angle between two straight lines; however, we’re not given equations for the straight lines. Instead, we’re given their direction cosines.
We need to give the measure of this angle to the nearest second. Let’s start by recalling how we determine the measure of the angle between two straight lines from their direction cosines. We know if we have two lines with direction cosines 𝑙 sub one, 𝑚 sub one, 𝑛 sub one and 𝑙 sub two, 𝑚 sub two, 𝑛 sub two, then the acute angle 𝜃 between the two lines will satisfy the equation cos of 𝜃 is equal to the absolute value of 𝑙 sub one times 𝑙 sub two plus 𝑚 sub one times 𝑚 sub two plus 𝑛 sub one times 𝑛 sub two. And since we’re given the direction cosines of the two lines, we can directly apply this formula.
First, we just find the values of 𝑙 sub one, 𝑚 sub one, 𝑛 sub one and 𝑙 sub two, 𝑚 sub two, and 𝑛 sub two from the question. Then all we need to do is substitute these values into our equation. We can then solve this equation for 𝜃. This then gives us the following equation. We need to solve this for 𝜃. So let’s start by simplifying the right-hand side of the equation.
First, let’s simplify the product of our terms. In the denominator of each of these terms, we have root two multiplied by root two. This will be equal to two. Since this is equal to two, we can simplify this. In our first term, we can cancel the shared factor of two in the numerator and denominator. And in our second term, we can also cancel the shared factor of two in the numerator and denominator. And then there are no more shared factors we can cancel, so we just need to evaluate each term separately. We get the absolute value of negative 15 over 169 minus 36 over 143 minus 27 over 112.
And now while we could evaluate the right-hand side of the equation exactly, it’s not necessary. In fact, we’ll just give an expansion. It’s the absolute value of negative 0.581 and this expansion continues. And now to take the absolute value of a negative number, we multiply it by negative one. We’ve shown the cos of 𝜃 is equal to 0.581 and this expansion continues.
We can now solve for the value of 𝜃 by taking the inverse cosine of both sides of the equation. Evaluating this where we make sure we use the exact values and our calculator is set two degrees mode gives us that 𝜃 is equal to 54.43 and this expansion continues degrees. So the measure of the angle between the two straight lines to one decimal place is 54.4 degrees. However, the question wants us to give our answer to the nearest second, so we’re going to need to convert this into degrees, minutes, and seconds. And to do this, we start by noting there’s 54 degrees in this angle. And this leaves us with a remaining 0.43 and this expansion continues degrees.
We want to know how many minutes are in this remaining angle. And to do this, we need to convert this angle into minutes. Since there are 60 minutes in a degree, we need to multiply the angle by 60. Calculating this where we make sure we use the exact values gives us 26.30 and this expansion continues minutes. So we can see we have 26 minutes and a remaining 0.3 and this expansion continues minutes.
We need to apply this process one final time to determine the remaining angle in seconds. To do this since there are 60 seconds in a minute, we need to multiply the remaining angle by 60. Using the exact value, we get 18.44 and this expansion continues seconds. And we want to give our answer to the nearest second. So we need to check the first decimal digit to determine if we need to round up or round down. This is less than five, so we need to round down, which then gives us our final answer. The measure of the angle between the two straight lines with the given direction cosines to the nearest second is 54 degrees, 26 minutes, and 18 seconds.
