Question

Solve the following equation on the interval [0°, 360°). Round answers to the nearest tenth. If there is no solution, indicate "No Solution."- 3sec?(x) - 11tan(x) = 3

Answer 1

Solve the equation on the interval [0°,360°)

`-3\sec ^2(x)-11\tan (x)=3`
`\begin{gathered} \mathrm{Subtract\: }3\mathrm{\: from\: both\: sides} \\ -3\sec ^2(x)-11\tan (x)-3=3-3 \\ -3\sec ^2(x)-11\tan (x)-3=0 \end{gathered}`

Rewrite using trigonometric identity

`\begin{gathered} 6-11\tan \mleft(x\mright)-3\tan ^2\mleft(x\mright)=0 \\ solve\text{ using substitution} \\ \tan (x)=y \end{gathered}`
`\begin{gathered} -6-11y-3y^2=0 \\ -6-2y-9y-3y^2=0 \\ -2(3+y)-3y(3+y)=0 \\ -2-3y=0,3+y=0 \\ -2=3y,y=-3 \\ y=-(2)/(3),y=-3 \end{gathered}`
`\begin{gathered} \tan \mleft(x\mright)=-3,\: \tan \mleft(x\mright)=-(2)/(3) \\ x=\arctan \mleft(-3\mright)+\pi n,\: x=\arctan \mleft(-(2)/(3)\mright)+\pi n \\ x=-71.56505^(\circ\: ),\: x=-33.69^(\circ\: ) \\ x=-71.6^0,x=-33.7^0\text{ (nearest tenth)} \end{gathered}`

Therefore the solution of the equation is

`x=-71.6^0,-33.7^0`