If $$ an heta=\frac{12}{5} $$ and $$ 90^{\circ}

Question

If $$ 	an 	heta=\frac{12}{5} $$ and $$ 90^{\circ}

UpStudy AI · Accepted Answer

Given that $$ \tan \theta = \frac{12}{5} $$ and $$ 90^{\circ} < \theta < 270^{\circ} $$, we can determine the value of $$ \frac{10}{\cos \theta} - \frac{5}{\cos \theta} $$ as follows:

1. **Determine the Quadrant:**
   - Since $$ 90^{\circ} < \theta < 270^{\circ} $$, $$ \theta $$ lies in either the second or third quadrant.
   - Given $$ \tan \theta = \frac{12}{5} $$ is positive, $$ \theta $$ must be in the **third quadrant** where both sine and cosine are negative.

2. **Find $$ \cos \theta $$:**
   - From $$ \tan \theta = \frac{12}{5} $$, consider a right triangle where the opposite side is 12 and the adjacent side is 5.
   - The hypotenuse is $$ \sqrt{5^2 + 12^2} = 13 $$.
   - In the third quadrant, $$ \cos \theta $$ is negative: $$ \cos \theta = -\frac{5}{13} $$.

3. **Compute the Expression:**
   $$
   \frac{10}{\cos \theta} - \frac{5}{\cos \theta} = \frac{10 - 5}{\cos \theta} = \frac{5}{\cos \theta}
   $$
   Substituting $$ \cos \theta $$:
   $$
   \frac{5}{-\frac{5}{13}} = 5 \times \left( -\frac{13}{5} \right) = -13
   $$

**Final Answer:** $$-13$$
The value of $$ \frac{10}{\cos \theta} - \frac{5}{\cos \theta} $$ is $$-13$$.

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