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 βˆ π΄π‘‚π΅ in standard position intersects with the unit circle at the point 𝐡 with coordinates (βˆ’π‘₯, βˆ’π‘₯). Find sin πœƒ.


Video Transcript

The terminal side of angle 𝐴𝑂𝐡 in standard position intersects with the unit circle at the point 𝐡 with coordinates negative π‘₯, negative π‘₯. Find the sine of πœƒ.

While we could answer this question with just the given information, I think it’s more helpful to sketch a picture. We’ll start with the circle that can be our unit circle. We sketch in the π‘₯-axis and the 𝑦-axis. We’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 𝐡 falls along the unit circle at point negative π‘₯, negative π‘₯. The terminal side of our angle intersects this point 𝐡. And that means we’re interested in this distance. To find the sine of this angle, we’ll need to remember some facts about the unit circle. We need to remember that 90 degrees in radians is equal to πœ‹ over two.

The first quarter of our turn is an angle of half πœ‹. The second quarter is another half πœ‹. One-half plus one-half equals an angle measure of πœ‹. Now, point negative π‘₯, negative π‘₯ is halfway between our next quarter turn. It is one-half of one-half. We’re turning πœ‹ over four, one-fourth πœ‹. And remember that our πœƒ, our angle measure, starts at line 𝑂𝐴 and goes to the terminal side 𝑂𝐡.

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

At this step you’ll need a computer or a calculator or a chart that lists out all of your sine functions, or perhaps you’ve memorized all the sine functions in the unit circle. That’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œ‹ over four is negative one over the square root of two.

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