Find approximate solutions for the equation in the interval $ -\pi

Question

Find approximate solutions for the equation in the interval $ -\pi<\mathbf{X}<\pi $. Round to three decimal places.
-$\tan x = 2 \csc x$
A) $- 1.144,1.144$
B) $- 1.571,1.571$
C) $1.144$
D) $- 0.785,0.785$

Quizplus · Accepted Answer

To solve the equation $\tan x = 2 \csc x$, we can express both trigonometric functions in terms of sine and cosine: $\frac{\sin x}{\cos x} = \frac{2}{\sin x}$. Multiplying both sides by $\sin x \cos x$ gives $\sin^2 x = 2 \cos x$. Using the Pythagorean identity $\sin^2 x = 1 - \cos^2 x$, we can substitute to get $1 - \cos^2 x = 2 \cos x$. Rearranging gives a quadratic equation in terms of $\cos x$, which can be solved using standard methods. The solutions to this equation in the given interval are approximately $-1.144$ and $1.144$, which corresponds to choice A.

Find Approximate Solutions for the Equation in the Interval −π<X<π -\pi<\mathbf{X}<\pi −π<X<π

Find Approximate Solutions for the Equation in the Interval $-\pi<\mathbf{X}<\pi$