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We propose a new attitude determination algorithm for a spacecraft in low earth orbits using magnetic sensors and rate sensors, which is applicable to a rotating spacecraft and does not require additional attitude sensors such as sun sensors. First, an attitude determination algorithm using only magnetic sensors is introduced presuming that the attitude change during the measurement period is very small. Then the algorithm is extended to combine magnetic sensors with rate sensors, where both the three-axis attitude and rate sensor biases are estimated. The estimation error is statistically analyzed. Numerical examples are given to validate the proposed algorithm and its accuracy evaluation.