If cos θ = 5/13, what is the value of tan \(\theta\) for 0 < θ < 90° ?
Answer Details
We can use the following trigonometric identity:
\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]
We are given that $\cos \theta = 5/13$. To find $\sin \theta$, we can use the Pythagorean identity:
\[\sin^2 \theta + \cos^2 \theta = 1\]
Substituting $\cos \theta = 5/13$, we get:
\[\sin^2 \theta + \left(\frac{5}{13}\right)^2 = 1\]
Simplifying, we get:
\[\sin^2 \theta = 1 - \frac{25}{169} = \frac{144}{169}\]
Taking the square root of both sides, we get:
\[\sin \theta = \frac{12}{13}\]
Substituting into the formula for $\tan \theta$, we get:
\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \boxed{\frac{12}{5}}\]