Video Transcript
Consider point 𝐴 with rectangular
coordinates negative four, six. Calculate the angle 𝜃 that line
segment 𝑂𝐴 makes with the positive 𝑥-axis. Give your answer in radians to two
decimal places.
In this question, we are given the
rectangular coordinates of a point 𝐴 and a sketch of point 𝐴 on a pair of
coordinate axes. We want to use the coordinates of
𝐴 and its sketch to determine the measure of the angle 𝜃 that line segment 𝑂𝐴
makes with the positive 𝑥-axis. To find the measure of this angle,
we can start by recalling that the sum of the measures of angles that form a
straight angle is 𝜋 radians. Therefore, the angle from line
segment 𝑂𝐴 to the negative 𝑥-axis will have measure 𝜋 minus 𝜃 radians.
By using the coordinates of point
𝐴, we can form a right triangle with side opposite the angle of length six and side
adjacent to the angle of length four. We can determine the unknown angle
by using trigonometry. We have that the tan of 𝜋 minus 𝜃
is equal to six over four. We can simplify six over four by
canceling the shared factor of two to get three over two. We can then rearrange for 𝜃 by
taking the inverse tangent of both sides of the equation. It is worth noting at this point
that we know that 𝜃 is obtuse since 𝐴 lies in the second quadrant. This means that 𝜋 minus 𝜃 will be
acute. Therefore, it is the unique
solution given by the inverse tan of three over two. We do not need to worry about any
other solutions.
We can now solve for 𝜃 by
rearranging the equation to make 𝜃 the subject and evaluating. We add 𝜃 to both sides of the
equation and subtract the inverse tan of three over two from both sides of the
equation. This gives us that 𝜃 is equal to
𝜋 minus the inverse tan of three over two. Evaluating this expression to two
decimal places gives us that 𝜃 is equal to 2.16 radians.
