Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problem 40

Chapter 2, Problem 40

Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3. cos θ > 0 , sec θ > 0

Verified step by step guidance

Recall the definitions and relationships between cosine and secant: \(\cos \theta\) is the cosine of the angle, and \(\sec \theta\) is its reciprocal, defined as \(\sec \theta = \frac{1}{\cos \theta}\).

Given that \(\cos \theta > 0\), this means the cosine value is positive. Cosine corresponds to the x-coordinate on the unit circle, so the angle \(\theta\) lies where the x-coordinate is positive.

Since \(\sec \theta = \frac{1}{\cos \theta}\), if \(\cos \theta > 0\), then \(\sec \theta\) must also be positive. This confirms the conditions are consistent and point to the same quadrants.

Recall the signs of cosine in each quadrant: cosine is positive in Quadrant I and Quadrant IV, and negative in Quadrant II and Quadrant III.

Therefore, the angle \(\theta\) must lie in Quadrant I or Quadrant IV, where both \(\cos \theta\) and \(\sec \theta\) are positive.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relationship Between Cosine and Secant

Cosine and secant are reciprocal trigonometric functions, meaning sec θ = 1/cos θ. Therefore, if cos θ > 0, sec θ must also be positive, and vice versa. Understanding this reciprocal relationship helps determine the sign of one function based on the other.

Signs of Trigonometric Functions in Quadrants

The signs of cosine and secant depend on the quadrant in which the angle θ lies. Cosine (and secant) is positive in the first and fourth quadrants, and negative in the second and third. Identifying the quadrant based on function signs is essential for solving the problem.

Quadrant Identification Using Inequalities

Given inequalities like cos θ > 0 and sec θ > 0, one can deduce the possible quadrants for θ by matching these conditions with the known sign patterns of trigonometric functions in each quadrant. This method is a fundamental approach to locating angles on the unit circle.

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Related Practice

Textbook Question

Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.

cos θ = ―5/8 , and θ is in quadrant III

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Find the measure of the smaller angle formed by the hands of a clock at the following times. 8:20

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Perform each calculation. See Example 3. 75° 15' + 83° 32'

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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

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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.

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Textbook Question

Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.

sec θ = ―√5 , and θ is in quadrant II

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