Suppose an arc of length s lies on the unit circle x² + y² = 1...

Question

Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.
s = 2.5
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s = 2.5

Accepted Answer

Welcome back. I am so glad you're here. We're told, consider an arc length of P on the unit circle X squared plus Y squared equals one. The starting and terminating points of the arc are 10 and XY respectively. With the help of a calculator determine the approximate coordinates of the point XY when P equals 3.5 round the answer to four decimal places. So here it's important to note that our P that's our arc length is 3.5. Now moving this to a different part of the screen or we can see the whole figure we have a drawing, we have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle, we have a unit circle on top of that. Every point on the unit circle is exactly one unit away from the origin. And then we have an angle alpha. The angle is in standard positions. The initial side is along the positive part of the X axis and then it's heading counter clockwise. The terminal side is in the third quadrant and it intercepts the unit circle at the coordinates. Xy, we're also told that X equals cosine P and Y equals sine P. We'll talk about that in a second. So we're trying to figure out what hour act and why values are. So we're going to recall a few things from previous lessons. We know that the arc length given by the variable S is equal to the radius of a circle multiplied by theta, the angle. But on the unit circle, the radius R is equal to one. So here our arc length S equals theta. Now we're told that our particular arc length is labeled P. So we can say that P equals theta. And so if we know that P equals 3.5 therefore theta equals 3.5 since P equals theta. So our angle is 3.5. Now another thing that we know from previous lessons, we know that the cosine of an angle on the unit circle. So here we'll say theta is equal to the X value at that angle. And the s of an angle on the unit circle is equal to the Y value of where that angle intercepts the unit circle. So we can figure out our X and our Y by finding the cosine and the sine of theta, theta, we know theta is 3.5 the same as P our arc length. So here we're just plugging this in and like it has been given to us the cosine of P. So 3.5 is going to be our X value and the sign of P 3.5 is going to equal our Y value. So here you just get your calculator, you plug in the cosine of 3.5 make sure that you are in radiance mode and you'll get something like negative 0.93645 that keeps going. But we were asked to round to four decimal places. So here our X value is going to be negative 0.936. That four is going to round up to a five. Now plugging in the sign of 3.5 into the calculator, you'll get something like negative 0.35078 and that keeps going rounding to the fourth decimal. The seven looks to the eight and rounds up. So this Y value will be negative 0.3508 and there's our XY coordinates. Well done. We catch you on the next one.

Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.
s = 2.5
<IMAGE>

Key Concepts

Unit Circle and Arc Length

Parametric Coordinates on the Unit Circle

Using a Calculator for Trigonometric Values

Watch next

Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.s = 2.5<IMAGE>

Key Concepts

Unit Circle and Arc Length

Parametric Coordinates on the Unit Circle

Using a Calculator for Trigonometric Values

Watch next

Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.
s = 2.5
<IMAGE>