Find all exact solutions on the interval $[0,2 \pi$ ). Look for opportunities to use trigonometric identities. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
\[
\sin ^{2}(x)-\cos ^{2}(x)-\sin (x)=0
\]
Answer
Final Answer: The solutions to the equation \(\sin ^{2}(x)-\cos ^{2}(x)-\sin (x)=0\) in the interval \([0,2 \pi)\) are \(x = 330^\circ\) and \(x = 90^\circ\). So, the final answer is \(\boxed{330^\circ, 90^\circ}\).
Step 1 ：Given the equation \(\sin ^{2}(x)-\cos ^{2}(x)-\sin (x)=0\).
Step 2 ：Use the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\) to rewrite the equation as \(\sin^2(x) - (1 - \sin^2(x)) - \sin(x) = 0\).
Step 3 ：Simplify the equation to get \(2\sin^2(x) - \sin(x) - 1 = 0\).
Step 4 ：This is a quadratic equation in terms of \(\sin(x)\), which can be solved using the quadratic formula to get the solutions \(-1/2\) and \(1\).
Step 5 ：Substitute these solutions back into the equation to find the corresponding values of \(x\) in the interval \([0, 2\pi)\). The solutions are \(x = -0.523598775598299 + 2\pi\) and \(x = 1.57079632679490\).
Step 6 ：Convert these solutions from radians to degrees to get \(x = 330^\circ\) and \(x = 90^\circ\).
Step 7 ：Final Answer: The solutions to the equation \(\sin ^{2}(x)-\cos ^{2}(x)-\sin (x)=0\) in the interval \([0,2 \pi)\) are \(x = 330^\circ\) and \(x = 90^\circ\). So, the final answer is \(\boxed{330^\circ, 90^\circ}\).
