The equation, with a restribtion on \( x \), is the lemminal side of an angle of in standard position. \[ 3 x+y=0, x \geq 0 \] ver 47 \[ \sin 0=-\frac{3 \sqrt{10}}{10} \] (simplify your answer Use integers of factons for any numbers in the expression. Type an exact answen, using radicats as needed. Rationaltye all denominators) B. The Anction is undafned. Selact the cored cholce balow and, if necessang, fill in the answer box to complate your choiee. A. \( 8080=\frac{\sqrt{10}}{10} \) (simplify your answer Use integers of factions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalise all denominators) B. The Anction is undafined. Select the comect cholee below and, if neeossany, fili in the answer box to complete your choiee. A. \( \tan 0=-3 \) (Simplity your answer. Use integers of factions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalise all denominators) B. The Annction is undefined. Select the correct choloe below and, if necessary, fill in the answer box to complete your choiees A. \( 2880= \) \( \square \) (Simplity your answer. Use inlegers of factions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalife all denominatons) B. The function is undefined. Clear all

Let's break down the problem step by step. ### Step 1: Analyze the given equation The equation given is: \[ 3x + y = 0 \] with the restriction \( x \geq 0 \). We can rearrange this equation to express \( y \) in terms of \( x \): \[ y = -3x \] ### Step 2: Determine the angle in standard position The line represented by the equation \( y = -3x \) has a slope of \(-3\). The angle \( \theta \) that this line makes with the positive x-axis can be found using the tangent function: \[ \tan(\theta) = \frac{y}{x} = -3 \] ### Step 3: Find the sine of the angle To find \( \sin(\theta) \), we can use the relationship between sine, cosine, and tangent. We know: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \] From the tangent value, we can set: \[ \sin(\theta) = -3k \quad \text{and} \quad \cos(\theta) = k \] for some positive \( k \). Using the Pythagorean identity: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] we substitute: \[ (-3k)^2 + k^2 = 1 \] which simplifies to: \[ 9k^2 + k^2 = 1 \implies 10k^2 = 1 \implies k^2 = \frac{1}{10} \implies k = \frac{1}{\sqrt{10}} \] Now substituting back to find \( \sin(\theta) \): \[ \sin(\theta) = -3k = -3 \cdot \frac{1}{\sqrt{10}} = -\frac{3}{\sqrt{10}} \] To rationalize the denominator: \[ \sin(\theta) = -\frac{3\sqrt{10}}{10} \] ### Step 4: Evaluate the choices 1. **For \( \sin(0) \)**: - The statement \( \sin(0) = -\frac{3\sqrt{10}}{10} \) is incorrect because \( \sin(0) = 0 \). Thus, the function is undefined. 2. **For \( \tan(0) \)**: - The statement \( \tan(0) = -3 \) is incorrect because \( \tan(0) = 0 \). Thus, the function is undefined. 3. **For the last choice**: - The statement \( 2880 = \square \) is unclear without context. However, if we assume it relates to the previous calculations, we can conclude that the function is undefined. ### Final Answers - For \( \sin(0) \): **B. The function is undefined.** - For \( \tan(0) \): **B. The function is undefined.** - For \( 2880 = \square \): **B. The function is undefined.**