Questions: Find θ for (0^circ leq theta<360^circ). [ textsec theta=1.765, cot theta<0 ] [ theta=square ]

Transcript text: Find $\theta$ for $0^{\circ} \leq \theta<360^{\circ}$. \[ \boldsymbol{\operatorname { s e c }} \theta=1.765, \cot \theta<0 \] \[ \theta=\square \]

Solution

Solution Steps

To solve for $\theta$ given $\sec \theta = 1.765$ and $\cot \theta < 0$, we need to consider the following:

$\sec \theta = 1.765$ implies $\cos \theta = \frac{1}{1.765}$.
$\cot \theta < 0$ indicates that $\theta$ is in either the second or fourth quadrant, where cosine is positive and cotangent is negative.
Calculate $\theta$ using the inverse cosine function and adjust for the appropriate quadrants.

Step 1: Calculate $\cos \theta$

Given that $\sec \theta = 1.765$, we can find $\cos \theta$ using the relationship: \[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{1.765} \approx 0.5666 \]

Step 2: Find $\theta$ in radians

Next, we calculate $\theta$ using the inverse cosine function: \[ \theta_{\text{rad}} = \cos^{-1}(0.5666) \approx 0.9685 \text{ radians} \]

Step 3: Convert $\theta$ to degrees

Converting $\theta$ from radians to degrees gives: \[ \theta_{\text{deg}} = \frac{180}{\pi} \cdot 0.9685 \approx 55.49 \text{ degrees} \]

Step 4: Determine possible angles based on quadrants

Since $\cot \theta < 0$, $\theta$ must be in the second or fourth quadrant. We calculate the angles:

Second quadrant: \[ \theta = 180^\circ - 55.49 \approx 124.51 \text{ degrees} \quad \Rightarrow \quad \text{rounded to } 125 \text{ degrees} \]
Fourth quadrant: \[ \theta = 360^\circ - 55.49 \approx 304.51 \text{ degrees} \quad \Rightarrow \quad \text{rounded to } 305 \text{ degrees} \]