Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―2√3 , 2)
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
Problem 40
Textbook Question
Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) II , y/x
Verified step by step guidance1
Recall that the ratio given is \( \frac{y}{x} \), where \(x\) and \(y\) are the coordinates of a point in the plane.
Identify the signs of \(x\) and \(y\) in Quadrant II. In this quadrant, \(x < 0\) (negative) and \(y > 0\) (positive).
Since \(y\) is positive and \(x\) is negative, the ratio \( \frac{y}{x} \) is a positive number divided by a negative number.
Dividing a positive number by a negative number results in a negative value, so \( \frac{y}{x} < 0 \) in Quadrant II.
Therefore, the ratio \( \frac{y}{x} \) is negative when the point \((x, y)\) lies in Quadrant II.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Coordinate Plane Quadrants
The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant II, x is negative and y is positive. Understanding the sign of coordinates in each quadrant helps determine the sign of ratios like y/x.
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Sign of Ratios in Different Quadrants
The sign of a ratio such as y/x depends on the signs of y and x individually. Since y is positive and x is negative in Quadrant II, the ratio y/x will be negative. This concept is crucial for evaluating trigonometric ratios based on point location.
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Distance from Origin (r = √(x² + y²))
The distance r from the origin to the point (x, y) is always positive and is calculated using the Pythagorean theorem. While r is not directly needed to find the sign of y/x, it is fundamental in defining trigonometric functions and understanding the point's position.
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