Solve \( \cos \left(\frac{\pi}{2}-\theta\right)+\sin \theta+0.6=0 \) for \( 0 \leq \theta<2 \pi \)

To solve the equation: \[ \cos\left(\frac{\pi}{2} - \theta\right) + \sin \theta + 0.6 = 0 \quad \text{for} \quad 0 \leq \theta < 2\pi, \] follow these steps: 1. **Simplify the Trigonometric Expression:** Use the co-function identity: \[ \cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta. \] Substituting this into the original equation gives: \[ \sin \theta + \sin \theta + 0.6 = 0 \quad \Rightarrow \quad 2\sin \theta + 0.6 = 0. \] 2. **Solve for \(\sin \theta\):** \[ 2\sin \theta + 0.6 = 0 \quad \Rightarrow \quad \sin \theta = -0.3. \] 3. **Find the General Solutions:** The general solutions for \(\sin \theta = -0.3\) within the interval \(0 \leq \theta < 2\pi\) are: \[ \theta = \pi + \arcsin(0.3) \quad \text{and} \quad \theta = 2\pi - \arcsin(0.3). \] 4. **Calculate the Specific Values:** Compute \(\arcsin(0.3)\): \[ \arcsin(0.3) \approx 0.3047 \text{ radians}. \] Thus, the solutions are: \[ \theta_1 = \pi + 0.3047 \approx 3.446 \text{ radians}, \] \[ \theta_2 = 2\pi - 0.3047 \approx 5.979 \text{ radians}. \] 5. **Final Answer:** The solutions within the interval \(0 \leq \theta < 2\pi\) are approximately: \[ \theta \approx 3.446 \text{ radians} \quad \text{and} \quad \theta \approx 5.979 \text{ radians}. \] **Summary of Solutions:** \[ \theta \approx 3.446 \ \text{radians} \quad \text{and} \quad \theta \approx 5.979 \ \text{radians}. \]