What is the general solution for 3sin^2(θ)+4cos^2(θ)=4 ?

The general solution for 3sin^2(θ)+4cos^2(θ)=4 is θ=2pin,θ=pi+2pin

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Popular >

3sin^2(θ)+4cos^2(θ)=4

Solution

3 sin^{2} (θ) + 4 cos^{2} (θ) = 4

Solution

θ = 2 πn, θ = π + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

3 sin^{2} (θ) + 4 cos^{2} (θ) = 4

Subtract

4

from both sides

3 sin^{2} (θ) + 4 cos^{2} (θ) - 4 = 0

Rewrite using trig identities

- 4 + 3 sin^{2} (θ) + 4 cos^{2} (θ)

Use the Pythagorean identity:

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

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

= - 4 + 3 sin^{2} (θ) + 4 (1 - sin^{2} (θ))

Simplify

- 4 + 3 sin^{2} (θ) + 4 (1 - sin^{2} (θ)) : - sin^{2} (θ)

- 4 + 3 sin^{2} (θ) + 4 (1 - sin^{2} (θ))

Expand

4 (1 - sin^{2} (θ)) : 4 - 4 sin^{2} (θ)

4 (1 - sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (θ)

= 4 \cdot 1 - 4 sin^{2} (θ)

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 sin^{2} (θ)

= - 4 + 3 sin^{2} (θ) + 4 - 4 sin^{2} (θ)

Simplify

- 4 + 3 sin^{2} (θ) + 4 - 4 sin^{2} (θ) : - sin^{2} (θ)

- 4 + 3 sin^{2} (θ) + 4 - 4 sin^{2} (θ)

Group like terms

= 3 sin^{2} (θ) - 4 sin^{2} (θ) - 4 + 4

Add similar elements:

3 sin^{2} (θ) - 4 sin^{2} (θ) = - sin^{2} (θ)

= - sin^{2} (θ) - 4 + 4

- 4 + 4 = 0

= - sin^{2} (θ)

= - sin^{2} (θ)

= - sin^{2} (θ)

- sin^{2} (θ) = 0

Divide both sides by

- 1

- sin^{2} (θ) = 0

Divide both sides by

- 1

- sin^{2} (θ) = 0

Divide both sides by

- 1

\frac{- sin ^{2} ( θ )}{- 1} = \frac{0}{- 1}

Simplify

sin^{2} (θ) = 0

sin^{2} (θ) = 0

Apply rule

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

sin (θ) = 0

General solutions for

sin (θ) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

Graph

Popular Examples

tan^2(x)-3tan(x)-2=0

tan^{2} (x) - 3 tan (x) - 2 = 0

sin(3x)+1=cos(3x)

sin (3 x) + 1 = cos (3 x)

6sin^2(x)=5

6 sin^{2} (x) = 5

tan(θ)= 46/33

tan (θ) = \frac{46}{33}

arcsin(y)= 1/2

arcsin (y) = \frac{1}{2}

Frequently Asked Questions (FAQ)

What is the general solution for 3sin^2(θ)+4cos^2(θ)=4 ?
The general solution for 3sin^2(θ)+4cos^2(θ)=4 is θ=2pin,θ=pi+2pin