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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 81
Chapter 4, Problem 81

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 = ―7.4
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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 \(\theta\) (in radians) measured from the positive x-axis as \((\cos(\theta), \sin(\theta))\).
Understand that the arc length \(s\) on the unit circle corresponds directly to the angle \(\theta\) in radians because the radius \(r\) of the unit circle is 1, and arc length \(s = r \times \theta = \theta\).
Given the arc length \(s = -7.4\), interpret this as the angle \(\theta = -7.4\) radians, which means the point is reached by moving clockwise from the point \((1, 0)\) since the angle is negative.
Calculate the coordinates \((x, y)\) by evaluating \(x = \cos(-7.4)\) and \(y = \sin(-7.4)\) using a calculator, ensuring the calculator is set to radian mode.
Round the resulting values of \(x\) and \(y\) to four decimal places to find the approximate coordinates of the point on the unit circle corresponding to the arc length \(s = -7.4\).

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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 in radians subtended by the arc at the center. Negative arc length indicates movement clockwise from the starting point (1, 0).
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Relationship Between Arc Length and Coordinates

For a point (x, y) on the unit circle, the coordinates can be found using trigonometric functions: x = cos(θ) and y = sin(θ), where θ is the angle in radians equal to the arc length s. This links the linear measure of the arc to angular position and coordinates.
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Using a Calculator for Trigonometric Values

Calculators can compute sine and cosine values for any real number input in radians. To find (x, y) for s = -7.4, input θ = -7.4 into cos(θ) and sin(θ) functions, ensuring the calculator is set to radian mode, to obtain approximate coordinates to four decimal places.
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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.

s = 2.5

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