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If the terminal side of an angle θ intersects the unit circle at (\( \frac{5}{13}, \frac{12}{13} \)), what...

1. If the terminal side of an angle θ intersects the unit circle at (\( \frac{5}{13}, \frac{12}{13} \)), what is \( \tan(θ) \)?
a. \( \frac{12}{5} \)
b. \( \frac{5}{12} \)
c. \( \frac{5}{13} \)
d. \( \frac{12}{13} \)
2. The terminal side of an angle θ in standard position in Quadrant I intersects the unit circle at \( (x, \frac{7}{25}) \). What is x?
a. \( \frac{24}{25} \)
b. \( \frac{25}{24} \)
c. \( \frac{18}{25} \)
d. \( \frac{7}{24} \)

Answer

  1. Answer: a. \( \frac{12}{5} \)
    Explanation: \( \tan(θ) = \frac{y}{x} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5} \).

  2. Answer: a. \( \frac{24}{25} \)
    Explanation: Since the point is on the unit circle, \( x^2 + \left(\frac{7}{25}\right)^2 = 1 \). Solving for \( x \), \( x^2 = 1 - \left(\frac{7}{25}\right)^2 = \frac{576}{625} \), so \( x = \frac{24}{25} \).

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