Video Transcript
Find the measure of the acute angle
between the two straight lines whose equations are 11𝑥 plus 10𝑦 minus 28 equals
zero and two 𝑥 plus 𝑦 plus 15 equals zero to the nearest second.
In this question, we are given the
equations of two straight lines in general form and asked to find the measure of the
acute angle between them to the nearest second. We can do this by recalling that
the acute angle 𝛼 between any two lines with slopes 𝑚 sub one and 𝑚 sub two is
given by the tan of 𝛼 equals the absolute value of 𝑚 sub one minus 𝑚 sub two over
one plus 𝑚 sub one times 𝑚 sub two.
Since both lines are given in the
general form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero for real constants 𝑎, 𝑏, and 𝑐,
we can recall that the slope 𝑚 of a line in general form is given by 𝑚 equals
negative 𝑎 over 𝑏 provided that 𝑏 is nonzero. We can use this to find the slopes
of the two lines.
For the first line, we have 𝑎
equals 11 and 𝑏 equals 10. So the slope 𝑚 sub one is given by
negative 11 over 10. For the second line, we have 𝑎
equals two and 𝑏 equals one. So the slope 𝑚 sub two is given by
negative two over one, which simplifies to give negative two.
We can now substitute these slopes
into the formula for the angle to obtain that the tan of 𝛼 equals the absolute
value of negative 11 over 10 minus negative two all over one plus negative 11 over
10 times negative two. We can evaluate the numerator and
denominator to get the absolute value of nine-tenths over sixteen fifths. We can then evaluate this
expression to see that the tan of 𝛼 is equal to nine over 32.
We can solve for 𝛼 by taking the
inverse tangent of both sides of the equation. Evaluating this expression using a
calculator set to degrees mode yields that 𝛼 is equal to 15.70 and this expansion
continues degrees. We want to give our answer in
degrees, minutes, and seconds. So we use the button on our
calculator to convert this into these units to obtain 𝛼 equals 15 degrees, 42
minutes, and 31.1 seconds to the nearest tenth of a second.
We can then round this to the
nearest second to find that the measure of the acute angle between the two straight
lines to the nearest second is 15 degrees, 42 minutes, and 31 seconds.
