Question Video: Evaluating a Trigonometric Function Using Information about the Terminal Side of the Angle and the Unit Circle Mathematics • 10th Grade

The terminal side of ∠𝐴𝑂𝐵 in standard position intersects the unit circle 𝑂 at the point 𝐵 with coordinates (3/√10, 𝑦), where 𝑦 > 0. Find the value of sin 𝐴𝑂𝐵.

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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.