Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problem 83

Chapter 4, Problem 83

For each value of s, use a calculator to find sin s and cos s, and then use the results to decide in which quadrant an angle of s radians lies.
s = 51

Verified step by step guidance

Understand that the angle given is in radians, and since \(s = 51\) radians is quite large, the first step is to reduce this angle to an equivalent angle between \(0\) and \(2\pi\) radians by finding its remainder when divided by \(2\pi\). This is done using the formula: \(s_{\text{reduced}} = s \mod 2\pi\).

Calculate \(\sin s_{\text{reduced}}\) and \(\cos s_{\text{reduced}}\) using a calculator. These values will help determine the quadrant of the angle because the signs of sine and cosine vary by quadrant.

Recall the sign rules for sine and cosine in each quadrant: - Quadrant I: \(\sin > 0\), \(\cos > 0\) - Quadrant II: \(\sin > 0\), \(\cos < 0\) - Quadrant III: \(\sin < 0\), \(\cos < 0\) - Quadrant IV: \(\sin < 0\), \(\cos > 0\)

Compare the signs of the calculated \(\sin s_{\text{reduced}}\) and \(\cos s_{\text{reduced}}\) to the sign rules above to identify the quadrant in which the angle lies.

Summarize the result by stating the quadrant based on the signs of sine and cosine for the reduced angle.

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Key Concepts

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

Unit Circle and Angle Measurement in Radians

The unit circle is a circle with radius 1 centered at the origin, used to define sine and cosine for all angles. Angles measured in radians correspond to arc lengths on this circle, where 2π radians equal 360°. Understanding how large angles like 51 radians wrap around the circle is essential to determine their position.

Sine and Cosine Functions

Sine and cosine are trigonometric functions that give the y- and x-coordinates, respectively, of a point on the unit circle corresponding to an angle. Their values range between -1 and 1 and vary periodically, which helps identify the angle's quadrant based on the signs of sin s and cos s.

Quadrants and Sign of Trigonometric Functions

The coordinate plane is divided into four quadrants, each with specific signs for sine and cosine: Quadrant I (+,+), II (+,-), III (-,-), and IV (-,+). By evaluating the signs of sin s and cos s, one can determine the quadrant where the angle s lies, even for angles greater than 2π radians.

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