# Trigonometric equation cos sin and power

The problem is $2\cos t - 3\sin^2t +2 = 0$. I get to $2\cos t -3\sin^2t =-2$ I think that I need to use a trigonometric identity like $\cos(x+y)$ and to divide $2\cos t -3\sin^2t$ with the $\sqrt{2^2+3^2}$

Do you know how to solve this? It should be $\sqrt{2^2 + 3^2}$

• do you want to solve $$2* cost - 3* sin^2t +2 = 0$$ ? – Khosrotash Jul 25 '15 at 18:30
• $$2* cost - 3* sin^2t +2 = 0\\$$ write $$sin^2 t=1-cos^2 t$$ then you have a quadratic equation ,solve for $cos t$ – Khosrotash Jul 25 '15 at 18:31

$$2 \cos t - 3 \sin^2t +2 = 0\\.$$ Write $$\sin^2 t=1-\cos^2 t,$$ then you have a quadratic equation. Solve for $\cos t$ $$2\cos t -3(1-\cos^2 t)+2=0\\3\cos ^2 t+2\cos t-3+2=0\\\cos t=-1,\;\frac{1}{3}.$$
We have, $$2\cos t-3\sin^2 t+2=0$$ $$\implies 2\cos t-3+3\cos^2 t+2=0$$ $$\implies 3\cos^2t+ 2\cos t-1=0$$ Factorizing the expression, we get $$(3\cos t-1)(\cos t+1)=0$$ $$\text{if}\ 3\cos t-1=0 \implies \cos t=\frac{1}{3}\implies \color{blue}{t=2n\pi\pm\cos^{-1}\left(\frac{1}{3}\right)}$$ $$\text{if}\ \cos t+1=0 \implies \cos t=-1 \implies \color{blue}{t=(2n+1)\pi}$$ Where, $n$ is any integer