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Popular >

sin^2(x)+5cos^2(x)=3

Solution

sin^{2} (x) + 5 cos^{2} (x) = 3

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (x) + 5 cos^{2} (x) = 3

Subtract

3

from both sides

sin^{2} (x) + 5 cos^{2} (x) - 3 = 0

Rewrite using trig identities

- 3 + sin^{2} (x) + 5 cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 3 + sin^{2} (x) + 5 (1 - sin^{2} (x))

Simplify

- 3 + sin^{2} (x) + 5 (1 - sin^{2} (x)) : - 4 sin^{2} (x) + 2

- 3 + sin^{2} (x) + 5 (1 - sin^{2} (x))

Expand

5 (1 - sin^{2} (x)) : 5 - 5 sin^{2} (x)

5 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 5, b = 1, c = sin^{2} (x)

= 5 \cdot 1 - 5 sin^{2} (x)

Multiply the numbers:

5 \cdot 1 = 5

= 5 - 5 sin^{2} (x)

= - 3 + sin^{2} (x) + 5 - 5 sin^{2} (x)

Simplify

- 3 + sin^{2} (x) + 5 - 5 sin^{2} (x) : - 4 sin^{2} (x) + 2

- 3 + sin^{2} (x) + 5 - 5 sin^{2} (x)

Group like terms

= sin^{2} (x) - 5 sin^{2} (x) - 3 + 5

Add similar elements:

sin^{2} (x) - 5 sin^{2} (x) = - 4 sin^{2} (x)

= - 4 sin^{2} (x) - 3 + 5

Add/Subtract the numbers:

- 3 + 5 = 2

= - 4 sin^{2} (x) + 2

= - 4 sin^{2} (x) + 2

= - 4 sin^{2} (x) + 2

2 - 4 sin^{2} (x) = 0

Solve by substitution

2 - 4 sin^{2} (x) = 0

Let:

sin (x) = u

2 - 4 u^{2} = 0

2 - 4 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

2 - 4 u^{2} = 0

Move

2

to the right side

2 - 4 u^{2} = 0

Subtract

2

from both sides

2 - 4 u^{2} - 2 = 0 - 2

Simplify

- 4 u^{2} = - 2

- 4 u^{2} = - 2

Divide both sides by

- 4

- 4 u^{2} = - 2

Divide both sides by

- 4

\frac{- 4 u ^{2}}{- 4} = \frac{- 2}{- 4}

Simplify

\frac{- 4 u ^{2}}{- 4} = \frac{- 2}{- 4}

Simplify

\frac{- 4 u ^{2}}{- 4} : u^{2}

\frac{- 4 u ^{2}}{- 4}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4 u ^{2}}{4}

Divide the numbers:

\frac{4}{4} = 1

= u^{2}

Simplify

\frac{- 2}{- 4} : \frac{1}{2}

\frac{- 2}{- 4}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

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}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{2}

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}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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