if $$ an ^{2} heta=7 $$ with $$ heta \in\left(-90^{\circ} ;-180^{\circ} ight) $$, determine gilue of the following without the use of a calmilat id with the gid of a cliagram

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if $$ 	an ^{2} 	heta=7 $$ with $$ 	heta \in\left(-90^{\circ} ;-180^{\circ}ight) $$, determine gilue of the following without the use of a calmilat id with the gid of a cliagram

UpStudy AI · Accepted Answer

To solve the problem where $$ \tan^2 \theta = 7 $$ and $$ \theta $$ is in the interval $$ (-90^\circ, -180^\circ) $$, we can follow these steps:

1. **Find $$ \tan \theta $$**:
   Since $$ \tan^2 \theta = 7 $$, we can take the square root to find $$ \tan \theta $$:
   $$
   \tan \theta = \pm \sqrt{7}
   $$

2. **Determine the sign of $$ \tan \theta $$**:
   The angle $$ \theta $$ is in the third quadrant (since it is between $$ -90^\circ $$ and $$ -180^\circ $$). In the third quadrant, the tangent function is positive. Therefore, we have:
   $$
   \tan \theta = \sqrt{7}
   $$

3. **Find $$ \theta $$**:
   To find $$ \theta $$, we can use the inverse tangent function:
   $$
   \theta = \tan^{-1}(\sqrt{7})
   $$
   However, since we are in the third quadrant, we need to adjust the angle:
   $$
   \theta = -180^\circ + \tan^{-1}(\sqrt{7})
   $$

4. **Calculate $$ \tan^{-1}(\sqrt{7}) $$**:
   While we cannot use a calculator, we can estimate $$ \tan^{-1}(\sqrt{7}) $$. We know that:
   - $$ \tan(45^\circ) = 1 $$
   - $$ \tan(60^\circ) \approx 1.732 $$
   - $$ \tan(70^\circ) \approx 2.747 $$

Since $$ \sqrt{7} \approx 2.64575 $$, we can estimate that:
   $$
   \tan^{-1}(\sqrt{7}) \text{ is between } 60^\circ \text{ and } 70^\circ
   $$

5. **Final Calculation**:
   Therefore, we can express $$ \theta $$ as:
   $$
   \theta \approx -180^\circ + 65^\circ = -115^\circ
   $$

Thus, the value of $$ \theta $$ is approximately $$ -115^\circ $$.
$$ \theta \approx -115^\circ $$

if \( \tan ^{2} \theta=7 \) with \( \theta \in\left(-90^{\circ} ;-180^{\circ}\right) \), determine gilue of the following without the use of a calmilat id with the gid of a cliagram

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