Question

Solve the following equation on the interval [0°, 360º). Round answers to the nearest tenth. If there is no solution, indicate "No Solution."2sec^2(x) - 13tan(x) = -13

Answer 1

Given

`2\sec ^2(x)-13\tan (x)=-13`

Add 13 to both sides

`\begin{gathered} 2\sec ^2(x)-13\tan (x)+13=-13+13 \\ 2\sec ^2(x)-13\tan (x)+13=0 \end{gathered}`

We have that

`\sec ^2(x)=1+\tan ^2(x)`

So, substitute in the above equation

`2(1+\tan ^2(x))-13\tan (x)+13=0`

Simplify

`\begin{gathered} 2+2\tan ^2(x)-13\tan (x)+13=0 \\ 15+2\tan ^2(x)-13\tan (x)=0 \end{gathered}`

Reordering the equation

`2\tan ^2(x)-13\tan (x)+15=0`

We get a quadratic equation, then solve by factoring

`(2\tan (x)-3)(\tan (x)-5)=0`

Separate the solutions

`\begin{gathered} 2\tan (x)-3=0 \\ 2\tan (x)-3+3=0+3 \\ 2\tan (x)=3 \\ (2\tan (x))/(2)=(3)/(2) \\ \tan (x)=(3)/(2) \end{gathered}`

And

`\begin{gathered} \tan (x)-5=0 \\ \tan (x)-5+5=0+5 \\ \tan (x)=5 \end{gathered}`

Next, solve for x for each solution

`\begin{gathered} \tan (x)=(3)/(2) \\ x=\tan ^(-1)((3)/(2)) \\ x=56.3 \end{gathered}`

And

`\begin{gathered} \tan (x)=5 \\ x=\tan ^(-1)(5) \\ x=78.7 \end{gathered}`

Answer:

x = 56.3° and x = 78.7°