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Ch. 3 - Radian Measure and The Unit Circle
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 3 - Radian Measure and The Unit CircleProblem 79
Chapter 4, Problem 79

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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Verified step by step guidance
1
Recall that the unit circle is defined by the equation \(x^{2} + y^{2} = 1\), and any point \((x, y)\) on the unit circle can be represented using the angle \(\theta\) measured from the positive x-axis as \(x = \cos(\theta)\) and \(y = \sin(\theta)\).
Understand that the arc length \(s\) on a unit circle is directly equal to the angle \(\theta\) in radians because the radius \(r = 1\), and arc length formula is \(s = r \theta\); thus, \(\theta = s\).
Given \(s = 2.5\), set \(\theta = 2.5\) radians. This angle corresponds to the position on the unit circle starting from \((1, 0)\) and moving counterclockwise by \(2.5\) radians.
Calculate the coordinates \((x, y)\) by evaluating \(x = \cos(2.5)\) and \(y = \sin(2.5)\) using a calculator, making sure your calculator is set to radians mode.
Round the values of \(x\) and \(y\) to four decimal places to get the approximate coordinates of the point on the unit circle at arc length \(s = 2.5\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Circle and Arc Length

The unit circle is a circle with radius 1 centered at the origin. The arc length s on the unit circle corresponds directly to the angle θ in radians subtended by the arc at the center. Since the radius is 1, the arc length s equals the angle θ, making it straightforward to relate linear distance along the circle to angular measure.
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Parametric Coordinates on the Unit Circle

Any point (x, y) on the unit circle can be represented as (cos θ, sin θ), where θ is the angle formed with the positive x-axis. Given an arc length s, which equals θ in radians, the coordinates of the endpoint of the arc starting at (1, 0) are (cos s, sin s). This allows conversion from arc length to Cartesian coordinates.
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Using a Calculator for Trigonometric Values

Calculators can compute sine and cosine values for given angles in radians. To find the coordinates (x, y) for an arc length s, input s as the angle in radians and calculate cos(s) and sin(s). Rounding these results to four decimal places provides the approximate coordinates on the unit circle.
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Textbook 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.

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