A point on the terminal side of an angle \( \theta \) in standard position is \( (-12,5) \). Find the exact value of each of the six trigonometric functions of \( \theta \).

To find the six trigonometric functions of the angle \( \theta \) whose terminal side passes through the point \( (-12, 5) \), we first need to determine the radius \( r \), which is the distance from the origin to the point \( (-12, 5) \). The radius can be calculated using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} \] where \( x = -12 \) and \( y = 5 \). Calculating \( r \): \[ r = \sqrt{(-12)^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] Now that we have \( r = 13 \), we can find the six trigonometric functions: 1. **Sine**: \[ \sin(\theta) = \frac{y}{r} = \frac{5}{13} \] 2. **Cosine**: \[ \cos(\theta) = \frac{x}{r} = \frac{-12}{13} \] 3. **Tangent**: \[ \tan(\theta) = \frac{y}{x} = \frac{5}{-12} = -\frac{5}{12} \] 4. **Cosecant** (the reciprocal of sine): \[ \csc(\theta) = \frac{r}{y} = \frac{13}{5} \] 5. **Secant** (the reciprocal of cosine): \[ \sec(\theta) = \frac{r}{x} = \frac{13}{-12} = -\frac{13}{12} \] 6. **Cotangent** (the reciprocal of tangent): \[ \cot(\theta) = \frac{x}{y} = \frac{-12}{5} \] Now, we can summarize the exact values of the six trigonometric functions of \( \theta \): \[ \sin(\theta) = \frac{5}{13}, \quad \cos(\theta) = -\frac{12}{13}, \quad \tan(\theta) = -\frac{5}{12} \] \[ \csc(\theta) = \frac{13}{5}, \quad \sec(\theta) = -\frac{13}{12}, \quad \cot(\theta) = -\frac{12}{5} \]