### Video Transcript

Determine to the nearest second the
measure of the angle between two straight lines having slopes of five and
one-quarter.

Knowing the slopes of our two
lines, thatโs ๐ one is equal to five and ๐ two is one-quarter, we can find the
acute angle ๐ผ between the lines using the formula tan ๐ผ or the tangent of ๐ผ is
the absolute value of ๐ one minus ๐ two over one plus ๐ one multiplied by ๐
two. In our case, this gives us the tan
of ๐ผ is the absolute value of five minus one over four all divided by one plus five
times one over four. The right-hand side evaluates to 19
divided by nine. So this is the tan of our angle
๐ผ. And now taking the inverse tangent
on both sides, we have ๐ผ is equal to the inverse tan of 19 over nine. And from our calculators, we find
to four decimal places that ๐ผ is 64.6538 degrees.

Weโre asked to find the angle to
the nearest second. And to do this, we recall that
there are 60 minutes in one degree and 60 seconds in one minute. We begin therefore by multiplying
the decimal part of our degrees by 60, and this gives us to four decimal places
39.2294 minutes. Now, multiplying the decimal part
of our minutes by 60 to four decimal places, this gives us 13.7666 seconds, which is
approximately 14 seconds. To the nearest second then, the
angle between our two straight lines is 64 degrees, 39 minutes, and 14 seconds.