Question

Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval (0,2x)Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."- sec(-x) = 5sec(x) + 1

Answer 1

For this problem, we are provided with the following expression:

`-\sec (-x)=5\sec (x)+1`

We need to solve it for x over the interval [0, 2pi).

We have:

`\sec (-x)=\sec (x)`

Therefore, we can replace the left side of the equation as shown:

`\begin{gathered} -\sec (x)=5\sec (x)+1 \\ \end{gathered}`

Now we need to isolate the sec(x) on the left side.

`\begin{gathered} -5\sec (x)-\sec (x)=1 \\ -6\sec (x)=1 \\ \sec (x)=-(1)/(6) \\ (1)/(\cos (x))=-(1)/(6) \\ \cos (x)=-6 \end{gathered}`

Now we can apply the arc cosine to determine the value of x.

`\begin{gathered} \arccos (\cos (x))=\arccos (-6) \\ x=\arccos (-6) \end{gathered}`

There are no real values for x that have a cosine equal to -6. Therefore, this problem has no real solution.