005 Sample Final A, Question 11

Question Solve the following equation in the interval $[0,2\pi )$

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

Step 1:
We need to get rid of the $\sin ^{2}(\theta )$ term. Since $\sin ^{2}(\theta )=1-\cos ^{2}(\theta )$, the equation becomes
$(1-\cos ^{2}(\theta ))-\cos ^{2}(\theta )=1+\cos(\theta )$.

Step 2:
If we simplify and move all the terms to the right hand side, we have $0=2\cos ^{2}(\theta )+\cos(\theta )$.

Step 3:
Now, factoring, we have $0=\cos(\theta )(2\cos(\theta )+1)$. Thus, either $\cos(\theta )=0$ or $2\cos(\theta )+1=0$.

Step 4:

Step 5:
The solutions to $\cos(\theta )=0$ in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 2\pi)} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=\frac{\pi}{2}} or

Final Answer: