Solve the equation on the interval $$ 0 \leq 0

Question

Solve the equation on the interval $$ 0 \leq 0<2 \pi $$ $$ \sec \frac{3 	heta}{2}=-2 $$

UpStudy AI · Accepted Answer

Solve the equation $$ \sec \frac{3 	heta}{2}=-2 $$.
Solve the equation by following steps:
- step0: Solve for $$	heta$$:
  $$\sec\left(\frac{3	heta }{2}ight)=-2$$
- step1: Find the domain:
  $$\sec\left(\frac{3	heta }{2}ight)=-2,	heta 
eq \frac{\pi }{3}+\frac{2k\pi }{3},k \in \mathbb{Z}$$
- step2: Use the inverse trigonometric function:
  $$\frac{3	heta }{2}=\operatorname{arcsec}\left(-2ight)$$
- step3: Calculate:
  $$\begin{align}&\frac{3	heta }{2}=\frac{2\pi }{3}\&\frac{3	heta }{2}=\frac{4\pi }{3}\end{align}$$
- step4: Add the period:
  $$\begin{align}&\frac{3	heta }{2}=\frac{2\pi }{3}+2k\pi ,k \in \mathbb{Z}\&\frac{3	heta }{2}=\frac{4\pi }{3}+2k\pi ,k \in \mathbb{Z}\end{align}$$
- step5: Calculate:
  $$\begin{align}&	heta =\frac{4\pi }{9}+\frac{4k\pi }{3},k \in \mathbb{Z}\&\frac{3	heta }{2}=\frac{4\pi }{3}+2k\pi ,k \in \mathbb{Z}\end{align}$$
- step6: Calculate:
  $$\begin{align}&	heta =\frac{4\pi }{9}+\frac{4k\pi }{3},k \in \mathbb{Z}\&	heta =\frac{8\pi }{9}+\frac{4k\pi }{3},k \in \mathbb{Z}\end{align}$$
- step7: Find the union:
  $$	heta =\left\{ \begin{array}{l}\frac{4\pi }{9}+\frac{4k\pi }{3}\\frac{8\pi }{9}+\frac{4k\pi }{3}\end{array}ight.,k \in \mathbb{Z}$$
- step8: Check if the solution is in the defined range:
  $$	heta =\left\{ \begin{array}{l}\frac{4\pi }{9}+\frac{4k\pi }{3}\\frac{8\pi }{9}+\frac{4k\pi }{3}\end{array}ight.,k \in \mathbb{Z},	heta 
eq \frac{\pi }{3}+\frac{2k\pi }{3},k \in \mathbb{Z}$$
- step9: Find the intersection:
  $$	heta =\left\{ \begin{array}{l}\frac{4\pi }{9}+\frac{4k\pi }{3}\\frac{8\pi }{9}+\frac{4k\pi }{3}\end{array}ight.,k \in \mathbb{Z}$$
The solution to the equation $$ \sec \frac{3 	heta}{2}=-2 $$ on the interval $$ 0 \leq 	heta < 2\pi $$ is $$ 	heta = \frac{4\pi}{9} + \frac{4k\pi}{3} $$ or $$ 	heta = \frac{8\pi}{9} + \frac{4k\pi}{3} $$, where $$ k $$ is an integer.
$$ 	heta = \frac{4\pi}{9} + \frac{4k\pi}{3} $$ or $$ 	heta = \frac{8\pi}{9} + \frac{4k\pi}{3} $$

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