Question

if sin x + √3 cos x = 2, then what us the value of x​

Answer 1

★ Given :

sin x + cos x = √2

✩ To find :

• X = ?

Solution :

given that

>>sin x + cos x = √2

squaring both sides

(sin x + cos x)^2 =(√2)^2

using identity (a+b)^2=a^2 + b^2 + 2ab

sin^2x + cos^2x+2 sinx coSX = 2

using identity sin^2 A+ cos^2 A = 1

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Math

5 points >

1 + 2 sinx cos x = 2

2 sin x cos x = 2 - 1

2 sin x COS X = 1

dividing by cos^2 x both sides

(2 sinx cosx)/cos^2 x =1 / cos^2x

2 tan x = sec^2 x

using identity 1+ tan^2 A = sec^2 A

2 tan x = 1 + tan^2 x

➡ tan^2 x - 2 tan x + 1 = 0

putting tan x = m

➡m^2 - 2 m+1=0

finding the roots of quadratic eqn

➡➡m^2 - m - m + 1 = 0

➡m (m-1)-1(m-1) = 0

(m-1) (m-1) = 0

from here we get

m = 1

since, tan x = m so, tan x = 1

tan x = 1

hence, x = 45° (Ans.)

Answer 2

Let’s see if we can avoid any radicals and any sort of ± reasoning about the signs.

sinx=3cosx just tells us tanx=3. Let’s do it in general and let t=tanx be the given and solve for y=sinxcosx.

tanx=sinxcosx⋅cosxcosx=sinxcosxcos2x

t=ycos2x

y=tcos2x=tsec2x=t1+tan2x=t1+t2

We have t=3

y=sinxcosx=31+32=310