# 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 β π΄ππ΅ in standard position intersects the unit circle π at the point π΅ with coordinates (3/β10, π¦), where π¦ > 0. Find the value of sin π΄ππ΅.

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### Video Transcript

The terminal side of angle π΄ππ΅ in standard position intersects the unit circle π at the point π΅ with coordinates three over root 10, π¦, where π¦ is greater than zero. Find the value of sin π΄ππ΅.

An angle is said to be in standard position if the vertex is at the origin and the initial side lies on the positive π₯-axis. Since angle π΄ππ΅ is in standard position and π΅ is not on the π₯-axis, the point π΄ must lie on the positive π₯-axis. We can therefore sketch angle π΄ππ΅ equals π on the unit circle. Since both the π₯- and π¦-coordinates are positive, point π΅ lies in the first quadrant.

We know that the π₯- and π¦-coordinates of a point on the unit circle given by an angle π are defined by π₯ equals cos π and π¦ equals sin π. The value of sin π΄ππ΅ is therefore equal to the value of the π¦-coordinate of point π΅.

By representing triangle π΄ππ΅ as a right triangle, we can find the value of π¦ by using the Pythagorean theorem. We have π¦ squared plus three over root 10 squared is equal to one squared. Simplifying this, our equation becomes π¦ squared plus nine over 10 equals one. Subtracting nine-tenths from both sides, we have π¦ squared is equal to one-tenth. Square rooting both sides, and since π¦ is greater than zero, we obtain π¦ is equal to one over root 10 units. The value of sin π΄ππ΅ is one over root 10.