Questions: Find θ for 0° ≤ θ<360°. tan θ=-1.556, cos θ>0 θ= ° (Round to two decimal places as needed.)

Transcript text: Find $\theta$ for $0^{\circ} \leq \theta<360^{\circ}$. \[ \begin{array}{l} \tan \theta=-1.556, \cos \theta>0 \\ \theta=\square^{\circ} \end{array} \] (Round to two decimal places as needed.)

Solution

Solution Steps

To find the angle $\theta$ given $\tan \theta = -1.556$ and $\cos \theta > 0$, we need to determine the quadrant in which $\theta$ lies. Since $\tan \theta$ is negative and $\cos \theta$ is positive, $\theta$ must be in the fourth quadrant. We can use the arctangent function to find the reference angle and then adjust it to find the angle in the correct quadrant.

Step 1: Determine the Reference Angle

Given that $ \tan \theta = -1.556 $, we first find the reference angle $ \theta_r $ using the arctangent function: \[ \theta_r = \tan^{-1}(|-1.556|) \approx 57.2722^\circ \]

Step 2: Identify the Quadrant

Since $ \tan \theta < 0 $ and $ \cos \theta > 0 $, we conclude that $ \theta $ is in the fourth quadrant.

Step 3: Calculate the Angle

To find the angle $ \theta $ in the fourth quadrant, we use the formula: \[ \theta = 360^\circ - \theta_r \approx 360^\circ - 57.2722^\circ \approx 302.7278^\circ \] Rounding to two decimal places, we have: \[ \theta \approx 302.73^\circ \]

Final Answer

$\boxed{\theta = 302.73^\circ}$

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