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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblem 40
Chapter 1, Problem 40

In Exercises 39–40, let θ be an angle in standard position. Name the quadrant in which θ lies.
tan θ > 0 and cos θ < 0

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1
Recall the signs of the trigonometric functions in each quadrant: In Quadrant I, all functions are positive; in Quadrant II, sine is positive while cosine and tangent are negative; in Quadrant III, tangent is positive while sine and cosine are negative; in Quadrant IV, cosine is positive while sine and tangent are negative.
Analyze the given conditions: \(\tan \theta > 0\) means tangent is positive, and \(\cos \theta < 0\) means cosine is negative.
From the sign chart, identify the quadrants where tangent is positive: Quadrant I and Quadrant III.
Identify the quadrants where cosine is negative: Quadrant II and Quadrant III.
Find the quadrant common to both conditions (tangent positive and cosine negative), which is Quadrant III.

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Key Concepts

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

Sign of Trigonometric Functions in Quadrants

The signs of sine, cosine, and tangent functions vary depending on the quadrant in which the angle lies. Knowing these sign patterns helps determine the quadrant based on given inequalities for trigonometric values.
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Quadratic Formula

Relationship Between Tangent, Sine, and Cosine

Tangent of an angle is the ratio of sine to cosine (tan θ = sin θ / cos θ). Understanding this relationship allows us to infer the signs of sine and cosine from the sign of tangent and vice versa.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Quadrant Identification Using Inequalities

By analyzing the given inequalities (tan θ > 0 and cos θ < 0), we can identify the quadrant where both conditions hold true. This involves matching the sign conditions to the known sign patterns of trig functions in each quadrant.
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Quadratic Formula
Related Practice
Textbook Question

In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places.

tan 32.7°

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Textbook Question

In Exercises 39–40, let θ be an angle in standard position. Name the quadrant in which θ lies.

tan θ > 0 and sec θ > 0

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In Exercises 35–40, convert each angle in radians to degrees. Round to two decimal places.

-4.8 radians

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In Exercises 35–60, find the reference angle for each angle.

355°

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Textbook Question

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.

cos t + cos(t + 1000𝜋) - tan t - tan(t + 999𝜋) - sin t + 4 sin(t - 1000𝜋)

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Textbook Question

In Exercises 35–40, convert each angle in radians to degrees. Round to two decimal places.

-5.2 radians