Respuesta :
Answer:
A
Step-by-step explanation:
just go to calculator and solve the equation. cos(3times10degrees-180degrees). you wil get the correct answer of -squareroot3/2.
The values of x that represent the solutions to the equation [tex]cos(3x-180)=-\frac{\sqrt{3} }{2}[/tex] are 10°, 110°, 130°
What is an equation?
"It is a mathematical statement which consists of equal symbol between two algebraic expressions."
What is the formula for cos(A - B)?
[tex]cos(A-B)=cos(A)cos(B)+sin(A)sin(B)[/tex]
For given question,
We have been given an equation [tex]cos(3x-180)=-\frac{\sqrt{3} }{2}[/tex]
Using the formula of cos(A- B),
[tex]\Rightarrow cos(3x-180)=-\frac{\sqrt{3} }{2}\\\\\Rightarrow cos(3x)cos(180)+sin(3x)sin(180)=-\frac{\sqrt{3} }{2}\\\\\Rightarrow cos(3x)\times (-1)+sin(3x)\times 0=-\frac{\sqrt{3} }{2}\\\\\Rightarrow -cos(3x)+0=-\frac{\sqrt{3} }{2}\\\\\ \Rightarrow -cos(3x)=-\frac{\sqrt{3} }{2}\\\\\Rightarrow cos(3x)=\frac{\sqrt{3} }{2}[/tex]
We know, [tex]cos(\frac{\pi}{6} )=\frac{\sqrt{3} }{2}[/tex]
[tex]\Rightarrow cos(3x)=\frac{\sqrt{3} }{2}\\\\\Rightarrow cos(3x)=cos(\frac{\pi}{6} )\\\\\Rightarrow 3x=\frac{\pi}{6}~~~or~~~3x=\frac{\pi}{6}+ 2\pi[/tex]
Case 1:
[tex]\Rightarrow 3x=\frac{\pi}{6}\\\\\Rightarrow x=\frac{\pi}{18}\\\\\Rightarrow x=10^{\circ}[/tex]
Case 2:
[tex]\Rightarrow 3x=\frac{\pi}{6}+2\pi\\\\\Rightarrow 3x=30^{\circ}+360^{\circ}\\\\\Rightarrow x=130^{\circ}[/tex]
Since 0 ≤ x < 180°, so we have two values of x.
x = 10° and 130°
Also, for x = 110°,
cos(3(110°) - 180°)
= cos (330° - 180°)
= cos (150°)
= cos(180° - 30°)
= - cos(30°)
= [tex]-\frac{\sqrt{3} }{2}[/tex]
Therefore, the values of x that represent the solutions to the equation [tex]cos(3x-180)=-\frac{\sqrt{3} }{2}[/tex] are 10°, 110°, 130°
Learn more about cos(A - B) here:
https://brainly.com/question/13094664
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