The equation, with a restriction on \( x \), is the terminal side of an angle \( \theta \) in standard position. \( \begin{array}{l}3 x+y=0, x \geq 0 \\ \text { Give the exact values of the six trigonometric functions of } \theta \text {. } \\ \text { Select the correct choice below and, If necessary, fill in the answer box to complete your choice. } \\ \text { A. } \sin \theta=\square \\ \text { (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators. } \\ \text { B. The function is undefined. }\end{array} . \begin{array}{l}\text { B }\end{array} \) ?
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To find the exact values of the six trigonometric functions of the angle \( \theta \), we start with the equation \( 3x + y = 0 \). This can be rearranged to \( y = -3x \). Since the restriction is \( x \geq 0 \), we can consider points on the line in the first and fourth quadrants. To find the values of the trigonometric functions, we can take a point on the line. Let's choose \( x = 1 \), then \( y = -3 \). The terminal point is \( (1, -3) \). The radius \( r \) can be found using the distance formula: \( r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-3)^2} = \sqrt{10} \). Now we can determine the trigonometric functions: - \( \sin \theta = \frac{y}{r} = \frac{-3}{\sqrt{10}} = -\frac{3\sqrt{10}}{10} \) - \( \cos \theta = \frac{x}{r} = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10} \) - \( \tan \theta = \frac{y}{x} = \frac{-3}{1} = -3 \) - \( \csc \theta = \frac{r}{y} = \frac{\sqrt{10}}{-3} = -\frac{\sqrt{10}}{3} \) - \( \sec \theta = \frac{r}{x} = \sqrt{10} \) - \( \cot \theta = \frac{x}{y} = \frac{1}{-3} = -\frac{1}{3} \) Thus, the exact values of \( \sin \theta \) and the other trigonometric functions can be confidently expressed in simplified form, following the instructions for proper formatting and rationalizing the denominators.
