# Question Video: Finding the Value of a Trigonometric Function given the Coordinates of the Point of Intersection of the Unit Circle with the Terminal Side of an Angle in Standard Position Mathematics

The terminal side of β π΄ππ΅ in standard position intersects with the unit circle at the point π΅ with coordinates (βπ₯, βπ₯). Find sin π.

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### Video Transcript

The terminal side of angle π΄ππ΅ in standard position intersects with the unit circle at the point π΅ with coordinates negative π₯, negative π₯. Find the sine of π.

While we could answer this question with just the given information, I think itβs more helpful to sketch a picture. Weβll start with the circle that can be our unit circle. We sketch in the π₯-axis and the π¦-axis. Weβre told that this angle is in the standard position. And that means that its vertex is at the origin and that its initial side is along the positive π₯-axis.

We also know that point π΅ falls along the unit circle at point negative π₯, negative π₯. The terminal side of our angle intersects this point π΅. And that means weβre interested in this distance. To find the sine of this angle, weβll need to remember some facts about the unit circle. We need to remember that 90 degrees in radians is equal to π over two.

The first quarter of our turn is an angle of half π. The second quarter is another half π. One-half plus one-half equals an angle measure of π. Now, point negative π₯, negative π₯ is halfway between our next quarter turn. It is one-half of one-half. Weβre turning π over four, one-fourth π. And remember that our π, our angle measure, starts at line ππ΄ and goes to the terminal side ππ΅.

It will be equal to π plus π over four. We can write one π as four over four π, and then we can add one-fourth π to that value. Our angle measure, our π, is equal to five π over four. Remember, weβre working in radians. We memorize that unit circle in radians. We can take this information, five π over four, and plug that in to a sine function, sine of five π over four.

At this step youβll need a computer or a calculator or a chart that lists out all of your sine functions, or perhaps youβve memorized all the sine functions in the unit circle. Thatβs not so hard, and it can be very useful! All of those methods should return negative one over the square root of two. Sine of five π over four is negative one over the square root of two.