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stealth61 [152]
2 years ago
9

Solve the equation on the interval [0,2pi] 5sec(x)+7=-3

Mathematics
2 answers:
vovangra [49]2 years ago
8 0

Answer:

Hello,

x\in \bigg\{\dfrac{2\pi}{3} ,\dfrac{4\pi}{3}\bigg\}\\

Step-by-step explanation:

5*sec(x)+7=-3\\\\sec(x)=\dfrac{-10}{5} \\\\\dfrac{1}{cos(x)} =-2\\\\cos(x)=-\dfrac{1}{2} \\\\x=120^o\ or\ x=240^o\\

Alex17521 [72]2 years ago
3 0

Answer:

\sf \boxed{\sf x = \dfrac{2\pi}{3},\dfrac{4\pi}{3}}

Step-by-step explanation:

A trigonometric equation is given to us , and we need to find the solutions of the equation within the interval [ 0,2π ]

The given equation is ,

\sf\longrightarrow 5 sec\  x +7 = -3

Add -7 to both sides ,

\sf\longrightarrow 5sec\ x = -10

Divide both sides by 5 ,

\sf\longrightarrow sec \ x =\dfrac{-10}{5}

Simplify ,

\sf\longrightarrow sec \ x =-2

Now solve for x ,

\sf\longrightarrow x = sec^{-1}(-2)

Simplify ,

\sf\longrightarrow x = \dfrac{2\pi}{3}

The secant function is <u>negative in the second and third quadrants. </u> Subtracting the reference angle from <u>2</u><u>π</u><u> </u> to find the solution in the third quadrant to find the solution second solution.

\sf\longrightarrow x = 2\pi -\dfrac{2\pi}{3}

Simplify ,

\sf\longrightarrow x = \dfrac{4\pi}{3}

Now here the period of sec x is <u>2</u><u>π</u> . Therefore ,

\sf\longrightarrow x = \dfrac{2\pi}{3} +2\pi n , \dfrac{4\pi}{3}+2\pi n , \textsf{ for any integer n } .

Therefore all the possible solutions are ,

\sf\longrightarrow \boxed{\blue{\sf x = \dfrac{2\pi}{3},\dfrac{4\pi}{3}}}

<u>Hence</u><u> the</u><u> </u><u>required</u><u> </u><u>answer</u><u> </u><u>is </u><u>2</u><u>π</u><u>/</u><u>3 </u><u>and </u><u>4</u><u>π</u><u>/</u><u>3.</u>

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