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

3. Unit Circle

Reference Angles

Trigonometry

3. Unit Circle

Reference Angles

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Problem 79

Textbook Question

Suppose θ is in the interval (90°, 180°). Find the sign of each of the following.sec(θ + 180°)

Verified step by step guidance

Recognize that \( \theta \) is in the second quadrant, where angles range from 90° to 180°.

Recall that the secant function, \( \sec(\theta) \), is the reciprocal of the cosine function, \( \cos(\theta) \).

Understand that \( \theta + 180° \) moves the angle to the third quadrant, where angles range from 180° to 270°.

In the third quadrant, the cosine of an angle is negative, so \( \cos(\theta + 180°) < 0 \).

Since \( \sec(\theta + 180°) = \frac{1}{\cos(\theta + 180°)} \), and \( \cos(\theta + 180°) \) is negative, \( \sec(\theta + 180°) \) is also negative.

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Key Concepts

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

Trigonometric Functions and Their Signs

Trigonometric functions such as secant, sine, and cosine have specific signs in different quadrants of the unit circle. In the interval (90°, 180°), which corresponds to the second quadrant, the sine function is positive while the cosine function is negative. Since secant is the reciprocal of cosine, its sign will also be negative in this quadrant.

Angle Addition in Trigonometry

The angle addition formula allows us to find the value of trigonometric functions for the sum of two angles. For sec(θ + 180°), we can use the property that sec(θ + 180°) = sec(θ) because the secant function has a periodicity of 360°. This means that the sign of sec(θ + 180°) will be the same as that of sec(θ).

Periodicity of Trigonometric Functions

Trigonometric functions exhibit periodic behavior, meaning their values repeat at regular intervals. For secant, the period is 360°, which implies that sec(θ + 180°) is equivalent to -sec(θ). Understanding this periodicity is crucial for determining the sign of sec(θ + 180°) based on the sign of sec(θ) in the specified interval.