The equation, with a restriction on $$ x $$, is the terminal side of an angle $$ heta $$ in standard position. $$ \begin{array}{l}3 x+y=0, x \geq 0 \ ext { Give the exact values of the six trigonometric functions of } heta ext {. } \ ext { Select the correct choice below and, If necessary, fill in the answer box to complete your choice. } \ ext { A. } \sin heta=\square \ ext { (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators. } \ ext { B. The function is undefined. }\end{array} . \begin{array}{l} ext { B }\end{array} $$ ?

Question

The equation, with a restriction on $$ x $$, is the terminal side of an angle $$ 	heta $$ in standard position. $$ \begin{array}{l}3 x+y=0, x \geq 0 \ 	ext { Give the exact values of the six trigonometric functions of } 	heta 	ext {. } \ 	ext { Select the correct choice below and, If necessary, fill in the answer box to complete your choice. } \ 	ext { A. } \sin 	heta=\square \ 	ext { (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators. } \ 	ext { B. The function is undefined. }\end{array} . \begin{array}{l}	ext { B }\end{array} $$ ?

UpStudy AI · Accepted Answer

To determine the exact values of the six trigonometric functions for the angle $$ \theta $$ whose terminal side is defined by the equation $$ 3x + y = 0 $$ with $$ x \geq 0 $$, let's follow these steps:

1. **Find a Point on the Terminal Side:**
   
   The equation $$ 3x + y = 0 $$ can be rewritten as $$ y = -3x $$. Since $$ x \geq 0 $$, let's choose $$ x = 1 $$ for simplicity:
   
   $$
   y = -3(1) = -3
   $$
   
   So, a point on the terminal side is $$ (1, -3) $$.

2. **Calculate the Distance $$ r $$ from the Origin to the Point:**
   
   $$
   r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
   $$

3. **Determine the Trigonometric Functions:**
   
   - **Sine ($$ \sin \theta $$):**
     
     $$
     \sin \theta = \frac{y}{r} = \frac{-3}{\sqrt{10}} = -\frac{3\sqrt{10}}{10}
     $$
   
   - **Cosine ($$ \cos \theta $$):**
     
     $$
     \cos \theta = \frac{x}{r} = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10}
     $$
   
   - **Tangent ($$ \tan \theta $$):**
     
     $$
     \tan \theta = \frac{y}{x} = \frac{-3}{1} = -3
     $$
   
   - **Cosecant ($$ \csc \theta $$):**
     
     $$
     \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3\sqrt{10}}{10}} = -\frac{\sqrt{10}}{3}
     $$
   
   - **Secant ($$ \sec \theta $$):**
     
     $$
     \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{10}}{10}} = \sqrt{10}
     $$
   
   - **Cotangent ($$ \cot \theta $$):**
     
     $$
     \cot \theta = \frac{1}{\tan \theta} = \frac{1}{-3} = -\frac{1}{3}
     $$

4. **Summary of the Six Trigonometric Functions:**

- $$ \sin \theta = -\dfrac{3\sqrt{10}}{10} $$
   - $$ \cos \theta = \dfrac{\sqrt{10}}{10} $$
   - $$ \tan \theta = -3 $$
   - $$ \csc \theta = -\dfrac{\sqrt{10}}{3} $$
   - $$ \sec \theta = \sqrt{10} $$
   - $$ \cot \theta = -\dfrac{1}{3} $$

**Therefore, the correct choice is A, with the trigonometric functions as calculated above.**
The trigonometric functions of $$ \theta $$ are: - $$ \sin \theta = -\dfrac{3\sqrt{10}}{10} $$ - $$ \cos \theta = \dfrac{\sqrt{10}}{10} $$ - $$ \tan \theta = -3 $$ - $$ \csc \theta = -\dfrac{\sqrt{10}}{3} $$ - $$ \sec \theta = \sqrt{10} $$ - $$ \cot \theta = -\dfrac{1}{3} $$ So, the answer is A.

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