Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. B. 60° 7. C. 82° 8. D. 30° 9. E. 38° 10. 480° F. 32°
Table of contents
- 0. Review of College Algebra4h 45m
- 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
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 guidance1
Recall the definition of the secant function: \(\sec(\alpha) = \frac{1}{\cos(\alpha)}\). To determine the sign of \(\sec(\theta + 180^\circ)\), we need to analyze the sign of \(\cos(\theta + 180^\circ)\).
Use the cosine angle addition identity for a shift by \(180^\circ\): \(\cos(\theta + 180^\circ) = -\cos(\theta)\).
Since \(\theta\) is in the interval \((90^\circ, 180^\circ)\), determine the sign of \(\cos(\theta)\) in this interval. Recall that cosine is negative in the second quadrant (between \(90^\circ\) and \(180^\circ\)).
Given that \(\cos(\theta)\) is negative in this interval, substitute back into the expression \(\cos(\theta + 180^\circ) = -\cos(\theta)\) to find its sign. Since \(\cos(\theta)\) is negative, \(-\cos(\theta)\) will be positive.
Finally, since \(\sec(\theta + 180^\circ) = \frac{1}{\cos(\theta + 180^\circ)}\), and \(\cos(\theta + 180^\circ)\) is positive, conclude that \(\sec(\theta + 180^\circ)\) is positive.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Function Periodicity
Trigonometric functions repeat their values in regular intervals called periods. For secant, which is the reciprocal of cosine, the period is 360°. This means sec(θ + 360°) = sec(θ), and understanding this helps simplify expressions involving angle shifts.
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Angle Addition and Quadrant Analysis
Adding angles shifts the position of the terminal side on the unit circle. Since θ is in (90°, 180°), adding 180° moves the angle to (270°, 360°). Knowing which quadrant the new angle lies in is essential to determine the sign of trigonometric functions.
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Sign of Secant Function in Different Quadrants
Secant is the reciprocal of cosine, so its sign depends on the cosine value. Cosine is negative in the second and third quadrants and positive in the first and fourth. Therefore, secant is negative where cosine is negative and positive where cosine is positive.
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