Video Transcript
The terminal side of angle
𝐴𝑂𝐵 in standard position intersects the unit circle 𝑂 at the point 𝐵 with
coordinates three over root 10, 𝑦, where 𝑦 is greater than zero. Find the value of sin
𝐴𝑂𝐵.
An angle is said to be in
standard position if the vertex is at the origin and the initial side lies on
the positive 𝑥-axis. Since angle 𝐴𝑂𝐵 is in
standard position and 𝐵 is not on the 𝑥-axis, the point 𝐴 must lie on the
positive 𝑥-axis. We can therefore sketch angle
𝐴𝑂𝐵 equals 𝜃 on the unit circle. Since both the 𝑥- and
𝑦-coordinates are positive, point 𝐵 lies in the first quadrant.
We know that the 𝑥- and
𝑦-coordinates of a point on the unit circle given by an angle 𝜃 are defined by
𝑥 equals cos 𝜃 and 𝑦 equals sin 𝜃. The value of sin 𝐴𝑂𝐵 is
therefore equal to the value of the 𝑦-coordinate of point 𝐵.
By representing triangle 𝐴𝑂𝐵
as a right triangle, we can find the value of 𝑦 by using the Pythagorean
theorem. We have 𝑦 squared plus three
over root 10 squared is equal to one squared. Simplifying this, our equation
becomes 𝑦 squared plus nine over 10 equals one. Subtracting nine-tenths from
both sides, we have 𝑦 squared is equal to one-tenth. Square rooting both sides, and
since 𝑦 is greater than zero, we obtain 𝑦 is equal to one over root 10
units. The value of sin 𝐴𝑂𝐵 is one
over root 10.
