Questions: Find the exact value of the trigonometric function of θ from the given information. cos θ = 5/13, θ in quadrant I, find tan θ tan θ = □ □ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution Steps

To find \(\tan \theta\) given \(\cos \theta = \frac{5}{13}\) and \(\theta\) in quadrant I, we can use the Pythagorean identity. First, we find \(\sin \theta\) using the identity \(\sin^2 \theta + \cos^2 \theta = 1\). Then, we use the definition of tangent, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

1. Use the Pythagorean identity to find \(\sin \theta\).
2. Calculate \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Given \(\cos \theta = \frac{5}{13}\), we can use the Pythagorean identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Substituting the value of \(\cos \theta\):

\[ \sin^2 \theta + \left(\frac{5}{13}\right)^2 = 1 \]

Calculating \(\left(\frac{5}{13}\right)^2\):

\[ \sin^2 \theta + \frac{25}{169} = 1 \]

Rearranging gives:

\[ \sin^2 \theta = 1 - \frac{25}{169} = \frac{169}{169} - \frac{25}{169} = \frac{144}{169} \]

Taking the square root (since \(\theta\) is in quadrant I, \(\sin \theta\) is positive):

\[ \sin \theta = \sqrt{\frac{144}{169}} = \frac{12}{13} \]

Now that we have both \(\sin \theta\) and \(\cos \theta\), we can find \(\tan \theta\) using the definition:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

Substituting the values:

\[ \tan \theta = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5} \]

Final Answer