Answers to Unit Circle Tutorials

This page provides complete answers and explanations to the questions from the tutorial Unit Circle and Trigonometric Functions: sin(x), cos(x), tan(x) .

1. Question:
   Is there a point on the unit circle that cannot have x- or y-coordinates? What are the domains of \( \sin(x) \) and \( \cos(x) \)?
   
   Answer:
   Every point on the unit circle has both an x-coordinate and a y-coordinate. These coordinates are defined as: \[ x = \cos(x), \quad y = \sin(x) \] Since an angle \(x\) can take any real value, both sine and cosine are defined for all real numbers. \[ \text{Domain of } \sin(x) = \text{Domain of } \cos(x) = (-\infty, +\infty) \]

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2. Question:
   Use the unit circle to determine the x-intercepts, maximums, and minimums of \( \sin(x) \) and \( \cos(x) \).
   
   Answer:
   Let \(k\) be any integer.

   x-intercepts:

   \[ \sin(x) = 0 \quad \text{when} \quad x = k\pi \] \[ \cos(x) = 0 \quad \text{when} \quad x = \frac{\pi}{2} + k\pi \]

   Maximum values:

   \[ \sin(x) = 1 \quad \text{at} \quad x = \frac{\pi}{2} + 2k\pi \] \[ \cos(x) = 1 \quad \text{at} \quad x = 2k\pi \]

   Minimum values:

   \[ \sin(x) = -1 \quad \text{at} \quad x = \frac{3\pi}{2} + 2k\pi \] \[ \cos(x) = -1 \quad \text{at} \quad x = \pi + 2k\pi \] These results follow directly from the positions of points on the unit circle.

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3. Question:
   Can the coordinates of a point on the unit circle be greater than 1 or less than -1? Explain the range of \( \sin(x) \) and \( \cos(x) \).
   
   Answer:
   The unit circle has radius 1, so every point lies exactly one unit from the origin. Therefore, neither the x-coordinate nor the y-coordinate can exceed 1 in absolute value. Since: \[ x = \cos(x), \quad y = \sin(x) \] their ranges are: \[ -1 \le \cos(x) \le 1 \] \[ -1 \le \sin(x) \le 1 \]

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4. Question:
   Explore the periodicity of \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \).
   
   Answer:
   When a point travels once around the unit circle, the angle increases by \(2\pi\). At this point, sine and cosine values repeat. \[ \text{Period of } \sin(x) = \text{Period of } \cos(x) = 2\pi \] The tangent function repeats after only half a revolution: \[ \text{Period of } \tan(x) = \pi \]

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5. Question:
   When is \( \tan(x) \) undefined? What is the domain of \( \tan(x) \)?
   
   Answer:
   Since: \[ \tan(x) = \frac{\sin(x)}{\cos(x)} \] tangent is undefined whenever \( \cos(x) = 0 \). This occurs at: \[ x = \frac{\pi}{2} + k\pi \] Therefore, the domain of \( \tan(x) \) is: \[ (-\infty, +\infty) \setminus \left\{ \frac{\pi}{2} + k\pi \right\} \]

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6. Question:
   Explain the asymptotic behavior of \( \tan(x) \). What is its range, and where are its x-intercepts between \(0\) and \(2\pi\)?
   
   Answer:
   Near the values where \( \cos(x) = 0 \), the denominator of \( \tan(x) \) approaches zero. As a result, the function increases or decreases without bound, creating vertical asymptotes. This leads to the range: \[ \text{Range of } \tan(x) = (-\infty, +\infty) \] The x-intercepts occur where \( \tan(x) = 0 \), which happens when \( \sin(x) = 0 \): \[ x = k\pi \] Between \(0\) and \(2\pi\), the intercepts are: \[ (0,0),\ (\pi,0),\ (2\pi,0) \]