詳解：\(f(\theta ) = {\cos ^2}\theta  + \sin \theta  - 1\)

\(= 1 - {\sin ^2}\theta  + \sin \theta  - 1 =  - {\sin ^2}\theta  + \sin \theta \)

\(=  - ({\sin ^2}\theta  - \sin \theta  + \frac{1}{4}) + \frac{1}{4}\)

\( =  - {(\sin \theta  - \frac{1}{2})^2} + \frac{1}{4}\)    最大\(\frac{1}{4}\)

或 \( - {\sin ^2}\theta  + \sin \theta  = \sin \theta (1 - \sin \theta )\)

使用算幾不等式

\(\frac{{\sin \theta  + \left( {1 - \sin \theta } \right)}}{2} \ge \sqrt {\sin \theta \left( {1 - \sin \theta } \right)} \)
\(\frac{1}{2} \ge \sqrt {\sin \theta \left( {1 - \sin \theta } \right)} \) \

\(\frac{1}{4} \ge \sin \theta \left( {1 - \sin \theta } \right)\)

故最大值為\(\frac{1}{4}\)