Questions: Find the exact value of each of the remaining trigonometric functions of θ. sin θ = 12/13, θ in Quadrant I sec θ =

Solution Steps

To find the exact value of the secant function given \(\sin \theta = \frac{12}{13}\) and \(\theta\) in Quadrant I, we can use the Pythagorean identity and the definition of secant.

1. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos \theta\).
2. Calculate \(\cos \theta\) using the given \(\sin \theta\).
3. Use the definition \(\sec \theta = \frac{1}{\cos \theta}\) to find \(\sec \theta\).

Given \(\sin \theta = \frac{12}{13}\), we use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting \(\sin \theta\): \[ \left(\frac{12}{13}\right)^2 + \cos^2 \theta = 1 \] \[ \frac{144}{169} + \cos^2 \theta = 1 \] \[ \cos^2 \theta = 1 - \frac{144}{169} \] \[ \cos^2 \theta = \frac{169}{169} - \frac{144}{169} \] \[ \cos^2 \theta = \frac{25}{169} \] \[ \cos \theta = \sqrt{\frac{25}{169}} \] \[ \cos \theta = \frac{5}{13} \]

The secant function is defined as: \[ \sec \theta = \frac{1}{\cos \theta} \] Substituting \(\cos \theta\): \[ \sec \theta = \frac{1}{\frac{5}{13}} \] \[ \sec \theta = \frac{13}{5} \]

Final Answer