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Popular >

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

Solution

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

Solution

x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn, x = - 0.46364 \dots + 2 πn, x = π + 0.46364 \dots + 2 πn

Degrees

x = 26.56505 \dots^{\circ} + 36 0^{\circ} n, x = 153.43494 \dots^{\circ} + 36 0^{\circ} n, x = - 26.56505 \dots^{\circ} + 36 0^{\circ} n, x = 206.56505 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

sin^{2} (x) + 3

from both sides

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

Rewrite using trig identities

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

Simplify

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

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

Expand

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

4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

= 4 \cdot 1 - 4 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

Add/Subtract the numbers:

- 3 + 4 = 1

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

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

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

1 - 5 sin^{2} (x) = 0

Solve by substitution

1 - 5 sin^{2} (x) = 0

Let:

sin (x) = u

1 - 5 u^{2} = 0

1 - 5 u^{2} = 0 : u = \frac{1}{5}, u = - \frac{1}{5}

1 - 5 u^{2} = 0

Move

1

to the right side

1 - 5 u^{2} = 0

Subtract

1

from both sides

1 - 5 u^{2} - 1 = 0 - 1

Simplify

- 5 u^{2} = - 1

- 5 u^{2} = - 1

Divide both sides by

- 5

- 5 u^{2} = - 1

Divide both sides by

- 5

\frac{- 5 u ^{2}}{- 5} = \frac{- 1}{- 5}

Simplify

u^{2} = \frac{1}{5}

u^{2} = \frac{1}{5}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{5}, u = - \frac{1}{5}

Substitute back

u = sin (x)

sin (x) = \frac{1}{5}, sin (x) = - \frac{1}{5}

sin (x) = \frac{1}{5}, sin (x) = - \frac{1}{5}

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

sin (x) = \frac{1}{5}

Apply trig inverse properties

sin (x) = \frac{1}{5}

General solutions for

sin (x) = \frac{1}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{1}{5}) + 2 πn, x = π - arcsin (\frac{1}{5}) + 2 πn

x = arcsin (\frac{1}{5}) + 2 πn, x = π - arcsin (\frac{1}{5}) + 2 πn

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

sin (x) = - \frac{1}{5}

Apply trig inverse properties

sin (x) = - \frac{1}{5}

General solutions for

sin (x) = - \frac{1}{5}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

Combine all the solutions

x = arcsin (\frac{1}{5}) + 2 πn, x = π - arcsin (\frac{1}{5}) + 2 πn, x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

Show solutions in decimal form

x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn, x = - 0.46364 \dots + 2 πn, x = π + 0.46364 \dots + 2 πn

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