Question

Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval [0, 21).Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."3sin?(x) + 13sin(x) = -12

Answer 1

No solution

Step-by-step explanation:

The equation

`3(\sin x)^2+13\sin x=-12`

looks very much like a quadratic equation. Therefore, for the moment we say that

`y=\sin x`

and write the above equation as

`\begin{gathered} 3y^2+13y=-12 \\ \Rightarrow3y^2+13y+12=0 \end{gathered}`

Using the quadratic formula, we find that the solutions to the above equation are given by

`y=\frac{-13\pm\sqrt[]{13^2-4(3)(12)}}{2\cdot3}`
`\begin{gathered} y=-(4)/(3) \\ y=-3 \end{gathered}`

Reminding ourselves that actually y was sin(x) gives