Question
Ł 5.7 .2 Test (CST): Trigonometry Question 10 of 25 Which function does not have a period of \( 2 \pi \) ? A. \( y=\sec x \) B. \( y=\csc x \) C. \( y=\cos x \) D. \( y=\tan x \)
Ask by Warren Rowe. in the United States
Jan 31,2025
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Answer
\( y = \tan x \) does not have a period of \( 2\pi \).
Solution
To determine which function does not have a period of \( 2\pi \), let's analyze the periodicity of each function:
1. **\( y = \sec x \)**:
- **Period**: \( 2\pi \)
- **Explanation**: Secant is the reciprocal of cosine. Since cosine has a period of \( 2\pi \), secant also shares this period.
2. **\( y = \csc x \)**:
- **Period**: \( 2\pi \)
- **Explanation**: Cosecant is the reciprocal of sine. Sine has a period of \( 2\pi \), so cosecant inherits the same period.
3. **\( y = \cos x \)**:
- **Period**: \( 2\pi \)
- **Explanation**: The cosine function naturally has a period of \( 2\pi \).
4. **\( y = \tan x \)**:
- **Period**: \( \pi \)
- **Explanation**: Tangent is the ratio of sine to cosine. It has a shorter period of \( \pi \) because it repeats its values every \( \pi \) radians, not every \( 2\pi \) radians.
**Conclusion**: Among the given options, **\( y = \tan x \)** does not have a period of \( 2\pi \). Instead, its period is \( \pi \).
**Answer**: D. \( y=\tan x \)
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Simplify this solution Beyond the Answer
Let’s shine a light on the periodic nature of these functions! The secant and cosecant functions, \( y = \sec x \) and \( y = \csc x \), both have a period of \( 2\pi \) just like the cosine function \( y = \cos x \). However, the tangent function, \( y = \tan x \), has a shorter period of \( \pi \), meaning it repeats itself twice within the interval of \( 2\pi \). As for a little problem-solving tip, always remember to check the definitions and properties of trigonometric functions; recognizing how they behave over intervals can be a simple way to identify their periods! It might help to sketch their graphs for clearer insight.
