Question

General solution of cos + sec =5/2 is​

Answer 1

Answer:

sectheta + 1/sectheta=5/2

(sec^2theta+1)/sectheta=5/2

2sec^2theta +2=5sectheta+2=0

sectheta=1/2 or 2

sectheta - costheta=3/2 or -3/2

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Answer 2

Answer:

Step-by-step explanation:

Sec(theta) + cos(theta) = 5/2 ,

Sec(theta) + 1/sec(theta) = 5/2 ,

(sec^2(theta)+1) / sec(theta) = 5/2 ,

2 sec^2(theta) + 2 = 5 sec(theta) ,

2 sec^2(theta) - 5 sec(theta) + 2 = 0 ,

Let sec(theta)=x ,

then,

2 x^2 - 5 x + 2 = 0 ,

2 x^2 - (4+1)x + 2 = 0 ,

2 x^2 - 4 x -x + 2 = 0 ,

2 x(x-2)-1(x-2)=0 ,

(2 x-1)(x-2)=0 ,

Either (2 x-1)=0 => x=1/2 => sec(theta)=1/2 [Since sec(theta)=x] ,

Or (x-2)=0 => x=2 => sec(theta,

Case 1: sec(theta) = 1/2 ,

Now we have Sec(theta) + cos(theta) = 5/2 ,

1/2 + cos(theta) = 5/2 ,

cos(theta) =(5/2 - 1/2) ,

cos(theta)=2 ,

then,

Sec(theta) - cos(theta) = (1/2 - 2) = -3/2 ,

Case 2: sec(theta) = 2 ,

Now we have Sec(theta) + cos(theta) = 5/2 ,

2 + cos(theta) = 5/2 ,

cos(theta) =(5/2 - 2) ,

cos(theta)=1/2 ,

then,

Sec(theta) - cos(theta) = (2 - 1/2) = 3/2 ,

therefore :

sec(theta) - cos(theta)=3/2 or-3/2 ,

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EXPLANATION(Shortcut):

Sec(theta) + cos(theta) = 5/2 ,

(Sec(theta) + cos(theta))^2 = (5/2)^2 ,

Sec(theta)^2 + cos(theta)^2 + 2*Sec(theta)*cos(theta) = 25/4 ,

Sec(theta)^2 + cos(theta)^2 + 2*Sec(theta)*(1/Sec(theta)) = 25/4 ,

Sec(theta)^2 + cos(theta)^2 + 2 = 25/4 ,

[Sec(theta)^2 + cos(theta)^2 + 2]- 4 = [25/4] - 4 [Subtracting both sides by 4] ,

Sec(theta)^2 + cos(theta)^2 - 2 = 9/4 ,

Sec(theta)^2 + cos(theta)^2 - 2*Sec(theta)*(1/Sec(theta)) = 9/4,

Sec(theta)^2 + cos(theta)^2 - 2*Sec(theta)*cos(theta) = 9/4 ,

(Sec(theta) - cos(theta))^2 = (3/2)^2 ,

Sec(theta) - cos(theta) = +-3/2.

Hope it helps you!!