Calculate \( \theta \) if \( 2 \sin \left(\theta+15^{\circ}\right)=1,462 \) and \( 0^{\circ} \leq \theta \leq 90^{\circ} \)

To solve the equation \( 2 \sin \left(\theta + 15^{\circ}\right) = 1,462 \), we first isolate the sine function. 1. Divide both sides by 2: \[ \sin \left(\theta + 15^{\circ}\right) = \frac{1,462}{2} \] \[ \sin \left(\theta + 15^{\circ}\right) = 731 \] 2. Since the sine function can only take values between -1 and 1, we see that \( 731 \) is outside this range. Therefore, there are no solutions for \( \theta \) in the interval \( 0^{\circ} \leq \theta \leq 90^{\circ} \). Thus, the conclusion is that there is no valid angle \( \theta \) that satisfies the equation \( 2 \sin \left(\theta + 15^{\circ}\right) = 1,462 \) within the specified range.