Video Transcript
Consider a windmill with the
blades of length one meter. The position of the top 𝑃 of a
given blade is given by the coordinates 𝑎, 𝑏, which depends upon the angle 𝜃
as shown. Express 𝑎 and 𝑏 as functions
of the measure of angle 𝜃 in radians. If angle 𝜃 at a certain time
is five-thirds 𝜋, what will it be after the blade has completed half a
rotation?
Because the length of the
blades are one meter and in our diagram, this represents the radius of the
windmill, we can use our knowledge of the unit circle to help us solve this
problem. The distance from the center of
the windmill to point 𝑃 would be one. And we’re told that the point
𝑃 is located at 𝑎, 𝑏. And so, we can create a right
triangle with the 𝑥-axis and say that its side lengths are 𝑎 and 𝑏,
respectively. We use the distance from the
terminal side to the 𝑥-axis to calculate the angle 𝜃. We know that in the unit
circle, we can represent the angle as a sine and cosine relationship. For our angle 𝜃, the opposite
side length would be 𝑏 and its hypotenuse is one meter, so we have sin of 𝜃
equals 𝑏 over one. And we can represent 𝑏 as sin
of 𝜃.
Similarly, if we look at
cosine, we end up with the side length 𝑎 over one, which means we can say that
the distance 𝑎 must be equal to the cos of the angle 𝜃. And without more information,
this is as far as we can go with these two functions. We can say that 𝑏 equals sin
of 𝜃 and 𝑎 equals cos of 𝜃.
Part two of our questions says
if the angle of 𝜃 at a certain time is five 𝜋 over three, what will it be
after the blade has completed a half a rotation? First of all, we’ve already
been told that we’re operating in radians, and so it might be helpful to label
our coordinate plane. Beginning at the 𝑥-axis, zero
radians, then 𝜋 over two radians, 𝜋 radians, three 𝜋 over two radians, and a
full turn, which is two 𝜋 radians. In a system like this, a full
turn is two 𝜋. And therefore, a half turn is
going to be equal to 𝜋 radians.
If we start with the angle 𝜃
five 𝜋 over three radians and we add a half a rotation, we’re adding 𝜋 to that
angle. And just like adding any
fractions, we need a common denominator. We can write 𝜋 as three 𝜋
over three. We add five 𝜋 plus three 𝜋 to
get eight 𝜋. And the denominator wouldn’t
change. The angle 𝜃 after a half
rotation would be eight 𝜋 over three.
