Question

Solve, finding all solutions in [0,271). 25. 4= 4 sin’ x+3 26. cos” x = 2 cos x 27. 1–3 tan” x = 0

I know the limit is only question per post, so if you could please just help me out with number 25! I havent done this in awhile and I need my memory refreshed with how to do these problems. Thank you!

Answer 1

(25 Given trigonometric equation is 4-4 sin's +3 - 1= 4 sinoc & sint oc = & Sintxe = 6 Siña = (sin II) 7 Sim x = Sint Am The principle value of sc is a = 7 - The general solution of sintx = sina is The general solection of given trigonometric eruation is 869 na I s n€ ZA If n=o then ac= 1 T/ = x= TG E [0,2T) and x=-ñ ¢[0,217 if na 1 then x = IT & T/ X= 7TH , 20= 11 - x=2 , =57 a If n=2 then se = 2T - 7/6 =X = 2017 II & [0, 27), 80 = 277 Acer ? - HIT € (0,271) The required solections of given equation on Co, 2T) are * = 50/70

Answer 2

SOLUTION :

25.

4 = 4 sin^2 (x) + 3 

=> 4 - 3 = 4 sin^2 (x)

=> sin^2 (x) = 1/4 

=> sin (x) = +/- 1/2 

So, in the interval [0, 2pi) :

When sin (x) = + 1/2 : 

 

Positive value of sin(x) occurs in 1st and 2nd quadrants.

x = pi/6 , (pi - pi/6)

=> x = pi/6, 5pi/6

When sin (x) = - 1/2 :

Negative value of sin(x) occurs in 3rd and 4th quadrants.

x = (pi + pi/6), (2pi - pi/6)

=> x = 7pi/6, 11pi/6  

So, when sin (x) = +1/2 , - 1/2 

x = pi/6, 5pi/6, 7pi/6, 11pi/6 (ANSWER).