Part 3 of 6 Points: 0 of 1 The equation, with a restriction on \( x \), is the terminal side of an angle \( \theta \) in standard position. \( 3 x+y=0, x \geq 0 \) Give the exact values of the six trigonometric functions of \( \theta \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \sin \theta=-\frac{3 \sqrt{10}}{10} \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed, Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \cos \theta=\frac{\sqrt{10}}{10} \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \tan \theta= \) \( \square \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined.

Let's determine the exact values of the six trigonometric functions for the angle \( \theta \) given the equation \( 3x + y = 0 \) with the restriction \( x \geq 0 \). ### Step 1: Determine the Relationship The equation \( 3x + y = 0 \) can be rewritten as: \[ y = -3x \] This represents a line with a slope of \(-3\). Since \( x \geq 0 \) and the slope is negative, the terminal side of the angle \( \theta \) lies in the **fourth quadrant**. ### Step 2: Find Coordinates and Radius Choose a convenient value for \( x \). Let \( x = 1 \): \[ y = -3(1) = -3 \] So, the point on the terminal side is \( (1, -3) \). Calculate the radius \( r \): \[ r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \] ### Step 3: Calculate Trigonometric Functions 1. **Sine (\( \sin \theta \))** \[ \sin \theta = \frac{y}{r} = \frac{-3}{\sqrt{10}} = -\frac{3\sqrt{10}}{10} \] **Choice A is correct:** \[ \sin \theta = -\frac{3\sqrt{10}}{10} \] 2. **Cosine (\( \cos \theta \))** \[ \cos \theta = \frac{x}{r} = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10} \] **Choice A is correct:** \[ \cos \theta = \frac{\sqrt{10}}{10} \] 3. **Tangent (\( \tan \theta \))** \[ \tan \theta = \frac{y}{x} = \frac{-3}{1} = -3 \] **Choice A is correct:** \[ \tan \theta = -3 \] ### Summary of Exact Values \[ \begin{align*} \sin \theta &= -\frac{3\sqrt{10}}{10} \\ \cos \theta &= \frac{\sqrt{10}}{10} \\ \tan \theta &= -3 \\ \csc \theta &= -\frac{\sqrt{10}}{3} \\ \sec \theta &= \sqrt{10} \\ \cot \theta &= -\frac{1}{3} \end{align*} \] For the parts you've provided: - **Sine (\( \sin \theta \))**: **A.** \( \sin \theta = -\frac{3\sqrt{10}}{10} \) - **Cosine (\( \cos \theta \))**: **A.** \( \cos \theta = \frac{\sqrt{10}}{10} \) - **Tangent (\( \tan \theta \))**: **A.** \( \tan \theta = -3 \)