Textbook Question
Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.
sec θ = 5/4 , and θ is in quadrant IV
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Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 2, Problem 44
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.) III , r/y
Verified step by step guidance1
Recall that the point (x, y) lies in Quadrant III, where both x and y coordinates are negative. This means x < 0 and y < 0 in this quadrant.
Remember the definition of r, which is the distance from the origin to the point (x, y), given by the formula \(r = \sqrt{x^2 + y^2}\). Since it is a distance, r is always positive regardless of the quadrant.
The ratio given is \(\frac{r}{y}\). Since r is positive and y is negative in Quadrant III, the numerator is positive and the denominator is negative.
When dividing a positive number by a negative number, the result is negative. Therefore, the ratio \(\frac{r}{y}\) will be negative in Quadrant III.
To confirm your understanding, sketch the coordinate plane, plot a point in Quadrant III, and visually verify the signs of x, y, and r, then consider the ratio \(\frac{r}{y}\).
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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 III, both x and y values are negative. Understanding the sign of coordinates in each quadrant helps determine the sign of expressions involving x and y.
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Quadratic Formula
Distance Formula and Radius r
The radius r is defined as r = √(x² + y²), representing the distance from the origin to the point (x, y). Since it is a distance, r is always positive regardless of the quadrant. This property is crucial when analyzing ratios involving r.
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6:36Quadratic Formula
Sign of Ratios Involving Coordinates
When evaluating the sign of a ratio like r/y, consider the signs of numerator and denominator separately. Since r is positive and y is negative in Quadrant III, the ratio r/y will be negative. This approach helps determine the positivity or negativity of trigonometric ratios.
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Related Practice
Textbook Question
Perform each calculation. See Example 3. 90° ― 51° 28'
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Textbook Question
Perform each calculation. See Example 3. 47° 29' ― 71° 18'
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Textbook Question
Determine whether each statement is possible or impossible. b. tan θ = 1.4
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Textbook Question
Determine whether each statement is possible or impossible. a. sec θ = ―2/3
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Textbook Question
Concept Check Classify each triangle as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.