What is the general solution for sin^2(x)+2cos^2(x)=1 ?

The general solution for sin^2(x)+2cos^2(x)=1 is x= pi/2+2pin,x=(3pi)/2+2pin

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sin^2(x)+2cos^2(x)=1

sin^{2} (x) + 2 cos^{2} (x) = 1

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (x) + 2 cos^{2} (x) = 1

Subtract

1

from both sides

sin^{2} (x) + 2 cos^{2} (x) - 1 = 0

Rewrite using trig identities

- 1 + sin^{2} (x) + 2 cos^{2} (x)

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

1 - sin^{2} (x) = cos^{2} (x)

= 2 cos^{2} (x) - cos^{2} (x)

Simplify

= cos^{2} (x)

cos^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

\frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} = 1

2 cos^{2} (x) = - 3 sin (x) cos (x)

4 cos^{2} (x) = sin^{2} (x) + 3

2 cos^{2} (x) - sin^{2} (x) - 2 sin (x) = 0

cos^{2} (x) + 5 cos (x) - 2 = 0

What is the general solution for sin^2(x)+2cos^2(x)=1 ?
The general solution for sin^2(x)+2cos^2(x)=1 is x= pi/2+2pin,x=(3pi)/2+2pin