Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

1. Measuring Angles

Angles in Standard Position

Trigonometry

1. Measuring Angles

Angles in Standard Position

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1:58 minutes

Problem 39

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

Verified step by step guidance

Step 1: Understand the properties of the tangent and secant functions.

Step 2: Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).

Step 3: Analyze the condition \( \tan \theta > 0 \). This implies that both \( \sin \theta \) and \( \cos \theta \) have the same sign.

Step 4: Analyze the condition \( \sec \theta > 0 \). This implies that \( \cos \theta > 0 \).

Step 5: Combine the conditions: Since \( \tan \theta > 0 \) and \( \sec \theta > 0 \), \( \theta \) must be in a quadrant where both \( \sin \theta \) and \( \cos \theta \) are positive. Determine which quadrant this is.

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Key Concepts

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

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants based on the signs of the x and y coordinates. Quadrant I has both x and y positive, Quadrant II has x negative and y positive, Quadrant III has both negative, and Quadrant IV has x positive and y negative. Understanding these quadrants is essential for determining the signs of trigonometric functions based on the angle's position.

Trigonometric Functions and Their Signs

Each trigonometric function has specific signs in different quadrants. For instance, tangent (tan) is positive in Quadrants I and III, while secant (sec), being the reciprocal of cosine, is positive in Quadrants I and IV. Knowing these relationships helps in identifying the quadrant where the angle lies based on the given conditions of the functions.

Understanding Tangent and Secant

Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle, while secant is the reciprocal of cosine, representing the ratio of the hypotenuse to the adjacent side. The conditions tan θ > 0 and sec θ > 0 indicate that both functions are positive, which restricts θ to Quadrant I, where both sine and cosine are positive.