Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problema 50

Capitolo 2, Problema 50

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.) I , r/y

Guida verificata passo dopo passo

Identify the quadrant given in the problem, which is Quadrant I. In this quadrant, both x and y coordinates are positive, so \(x > 0\) and \(y > 0\).

Recall the formula for \(r\), the distance from the origin to the point \((x, y)\): \(r = \sqrt{x^2 + y^2}\). Since \(x^2\) and \(y^2\) are always non-negative, \(r\) is always positive.

Analyze the ratio given: \(\frac{r}{y}\). Since \(r > 0\) and \(y > 0\) in Quadrant I, both numerator and denominator are positive.

Because both numerator and denominator are positive, the ratio \(\frac{r}{y}\) must be positive.

To confirm your understanding, sketch the coordinate plane, plot a point in Quadrant I, and visually verify that \(r\) and \(y\) are positive, reinforcing why the ratio is positive.

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Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant I, both x and y are positive, which affects the sign of ratios involving these values.

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Percorso guidato

6:36

Quadratic Formula

Distance from Origin (r)

The distance r from the origin to a point (x, y) is given by r = √(x² + y²), which is always positive. This value represents the radius in polar coordinates and is crucial for understanding ratios involving r.

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Percorso guidato

6:58

Convert Points from Rectangular to Polar

Sign of Ratios Involving Coordinates

The sign of a ratio like r/y depends on the signs of numerator and denominator. Since r is always positive, the sign of r/y depends solely on y's sign, which varies by quadrant, helping determine if the ratio is positive or negative.

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Percorso guidato

05:32

Intro to Polar Coordinates

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Domanda del libro di testo

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.) I , y/r

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Solve each problem. Height of a Lunar Peak The lunar mountain peak Huygens has a height of 21,000 ft. The shadow of Huygens on a photograph was 2.8 mm, while the nearby mountain Bradley had a shadow of 1.8 mm on the same photograph. Calculate the height of Bradley. (Data from Webb, T., Celestial Objects for Common Telescopes, Dover Publications.)

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Perform each calculation. See Example 3. 90° ― 36° 18' 47"

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Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3.

csc θ > 0 , cot θ > 0

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