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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
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Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- 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
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 50
Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3.
tan θ < 0 , cot θ < 0
Guida verificata passo dopo passo1
Recall the definitions and signs of the trigonometric functions in each quadrant. The tangent function \(\tan \theta\) is positive in Quadrants I and III, and negative in Quadrants II and IV. The cotangent function \(\cot \theta\) has the same sign as tangent because \(\cot \theta = \frac{1}{\tan \theta}\), so it is also positive in Quadrants I and III, and negative in Quadrants II and IV.
Analyze the given inequalities: \(\tan \theta < 0\) means \(\theta\) lies in a quadrant where tangent is negative, which are Quadrants II and IV.
Similarly, \(\cot \theta < 0\) means cotangent is negative, so \(\theta\) must be in a quadrant where cotangent is negative, which are also Quadrants II and IV.
Since both \(\tan \theta\) and \(\cot \theta\) are negative, the angle \(\theta\) must be in the intersection of the quadrants where both are negative, which are Quadrants II and IV.
Therefore, the possible quadrants for \(\theta\) are Quadrant II and Quadrant IV.
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Signs of Trigonometric Functions in Quadrants
The signs of sine, cosine, tangent, and cotangent vary depending on the quadrant of the angle. Tangent and cotangent are positive or negative based on the signs of sine and cosine, since tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Understanding which quadrants yield positive or negative values for these functions is essential to determine the angle's location.
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Percorso guidato
6:36
Quadratic Formula
Relationship Between Tangent and Cotangent
Tangent and cotangent are reciprocal functions: tan θ = 1 / cot θ and cot θ = 1 / tan θ. Their signs are linked but can differ depending on the quadrant. Recognizing that both tan θ and cot θ depend on sine and cosine helps analyze their signs simultaneously to narrow down possible quadrants.
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Percorso guidato
5:37Introduction to Cotangent Graph
Quadrant Identification Using Inequalities
Given inequalities like tan θ < 0 and cot θ < 0, one must use the sign rules of trigonometric functions to identify which quadrants satisfy these conditions. Since tangent and cotangent change signs in specific quadrants, analyzing these inequalities helps pinpoint the possible quadrants where the angle θ lies.
Video consigliato:
Percorso guidato
6:36Quadratic Formula
Pratica correlata
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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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Domanda del libro di testo
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Domanda del libro di testo
Perform each calculation. See Example 3. 90° ― 17° 13'
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