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. (0, ―3)
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 34
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.) IV , x/y
Verified step by step guidance1
Recall that the point (x, y) lies in Quadrant IV. In this quadrant, the x-coordinate is positive and the y-coordinate is negative.
The ratio given is \( \frac{x}{y} \). Since \( x > 0 \) and \( y < 0 \) in Quadrant IV, the numerator is positive and the denominator is negative.
A positive number divided by a negative number results in a negative value.
Therefore, the ratio \( \frac{x}{y} \) in Quadrant IV is negative.
To visualize this, sketch the coordinate plane, mark Quadrant IV, and plot a point with positive x and negative y to see why the ratio is negative.
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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 sign conventions for x and y coordinates. In Quadrant IV, x is positive and y is negative. Understanding these sign rules is essential for determining the sign of ratios involving x and y.
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Sign of Ratios in Different Quadrants
The sign of a ratio like x/y depends on the signs of numerator and denominator. Since x and y have known signs in each quadrant, the ratio's sign can be deduced by dividing their signs. For example, in Quadrant IV, x/y is positive divided by negative, resulting in a negative ratio.
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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 itself is positive, it helps relate x and y to trigonometric functions and confirms the point's position relative to the origin.
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