Find $$ \sec ( heta) $$ if $$ \cot ( heta)=\frac{8}{15} $$ and $$ \cos ( heta)

Question

Find $$ \sec (	heta) $$ if $$ \cot (	heta)=\frac{8}{15} $$ and $$ \cos (	heta)<0 $$. Answer exactly. $$ \sec (	heta)=\square $$

UpStudy AI · Accepted Answer

Given that $$ \cot(\theta) = \frac{8}{15} $$ and $$ \cos(\theta) < 0 $$, we can determine $$ \sec(\theta) $$ as follows:

1. **Understanding the Ratios:**
   - $$ \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{8}{15} $$.
   - This suggests that in a right triangle, the adjacent side is 8 units and the opposite side is 15 units.

2. **Determining the Hypotenuse:**
   - Using the Pythagorean theorem: 
     $$
     \text{hypotenuse} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17.
     $$

3. **Considering the Quadrant:**
   - $$ \cos(\theta) < 0 $$ implies that $$ \theta $$ is in either the second or third quadrant.
   - Since $$ \cot(\theta) = \frac{8}{15} > 0 $$, and cotangent is positive only in the first and third quadrants, $$ \theta $$ must be in the **third quadrant**.
   - In the third quadrant, both adjacent and opposite sides are negative.

4. **Calculating $$ \cos(\theta) $$:**
   - $$ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-8}{17} $$.

5. **Finding $$ \sec(\theta) $$:**
   - $$ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{8}{17}} = -\frac{17}{8} $$.

**Answer:** $$ \sec (\theta) = -\dfrac{17}{8} $$
$$ \sec (\theta) = -\dfrac{17}{8} $$

Find if and . Answer exactly.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Trigonometry Questions