solve the following equation for all values of x: sin^2x+sinxcosx

Question
solve the following equation for all values of x: $$\displaystyle{{\sin}^{{2}}{x}}+{\sin{{x}}}{\cos{{x}}}$$

2021-03-09
$$\displaystyle{\sin{{\left({x}\right)}}}{\left({\sin{{\left({x}\right)}}}+{\cos{{\left({x}\right)}}}\right)}$$. If this expression equals zero, $$\displaystyle{\sin{{\left({x}\right)}}}={0}$$ or $$\displaystyle{\sin{{\left({x}\right)}}}=-{\cos{{\left({x}\right)}}}$$, so $$\displaystyle{\tan{{\left({x}\right)}}}=-{1}.$$
Solutions: $$\displaystyle{\sin{{\left({x}\right)}}}={0}:{x}={n}{\left(\pi\right)},{\tan{{\left({x}\right)}}}=-{1}:{x}={\left({4}{n}-{1}\right)}\frac{{\pi}}{{4}}$$ where n is an integer. x is in radians. To convert to degrees put $$\displaystyle{\left(\pi\right)}={180}$$: 0, 180, 360, ..., 135, 315, ... for example.
$$\displaystyle{\cos{{\left({2}{x}\right)}}}={1}-{2}{{\sin}^{{2}}{\left({x}\right)}}$$, so $$\displaystyle{{\sin}^{{2}}{\left({x}\right)}}=\frac{{{1}-{\cos{{\left({2}{x}\right)}}}}}{{2}}$$
$$\displaystyle{\sin{{\left({2}{x}\right)}}}={2}{\sin{{\left({x}\right)}}}{\cos{{\left({x}\right)}}}$$, so $$\displaystyle{\sin{{\left({x}\right)}}}{\cos{{\left({x}\right)}}}={\left(\frac{{1}}{{2}}\right)}{\sin{{\left({2}{x}\right)}}}$$
$$\displaystyle{{\sin}^{{2}}{\left({x}\right)}}+{\sin{{\left({x}\right)}}}{\cos{{\left({x}\right)}}}={\left(\frac{{1}}{{2}}\right)}{\left({1}-{\cos{{\left({2}{x}\right)}}}+{\sin{{\left({2}{x}\right)}}}\right)}$$. Above solutions apply.

Relevant Questions

Solve $$\displaystyle\sqrt{{3}}{\csc{{2}}}{x}={2}$$ for all values of x.
Solve the equation
$$\displaystyle{\sin{{\left({x}°-{20}°\right)}}}={\cos{{42}}}°$$ for x, where 0 < x < 90
How to prove the following:
$$\displaystyle{{\tan}^{{2}}{x}}+{1}+{\tan{{x}}}{\sec{{x}}}={1}+\frac{{\sin{{x}}}}{{{\cos}^{{2}}{x}}}$$
Solve the equation on the interval [0,2pi] $$\sin{{\left({x}+\frac{\pi}{{4}}\right)}}+ \sin{{\left({x}-\frac{\pi}{{4}}\right)}}={1}$$
Solve the equation $$\frac{1-\sin x}{1+\sin x}=(\sec x-\tan x)^{2}$$
Solve the equation $$\tan x \sec x \sin x= \tan^{2}x$$
Solve the equation $$\csc x - \sin x = \cot x \cos x$$
Solve $$4(\sin^{6}x+\cos^{6}x)=4-3\sin^{2} 2x$$