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Popular >

48sin^2(x)=48-24cos(x)

Solution

48 sin^{2} (x) = 48 - 24 cos (x)

Solution

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

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

48 sin^{2} (x) = 48 - 24 cos (x)

Subtract

48 - 24 cos (x)

from both sides

48 sin^{2} (x) - 48 + 24 cos (x) = 0

Rewrite using trig identities

- 48 + 24 cos (x) + 48 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 48 + 24 cos (x) + 48 (1 - cos^{2} (x))

Simplify

- 48 + 24 cos (x) + 48 (1 - cos^{2} (x)) : 24 cos (x) - 48 cos^{2} (x)

- 48 + 24 cos (x) + 48 (1 - cos^{2} (x))

Expand

48 (1 - cos^{2} (x)) : 48 - 48 cos^{2} (x)

48 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 48, b = 1, c = cos^{2} (x)

= 48 \cdot 1 - 48 cos^{2} (x)

Multiply the numbers:

48 \cdot 1 = 48

= 48 - 48 cos^{2} (x)

= - 48 + 24 cos (x) + 48 - 48 cos^{2} (x)

Simplify

- 48 + 24 cos (x) + 48 - 48 cos^{2} (x) : 24 cos (x) - 48 cos^{2} (x)

- 48 + 24 cos (x) + 48 - 48 cos^{2} (x)

Group like terms

= 24 cos (x) - 48 cos^{2} (x) - 48 + 48

- 48 + 48 = 0

= 24 cos (x) - 48 cos^{2} (x)

= 24 cos (x) - 48 cos^{2} (x)

= 24 cos (x) - 48 cos^{2} (x)

24 cos (x) - 48 cos^{2} (x) = 0

Solve by substitution

24 cos (x) - 48 cos^{2} (x) = 0

Let:

cos (x) = u

24 u - 48 u^{2} = 0

24 u - 48 u^{2} = 0 : u = 0, u = \frac{1}{2}

24 u - 48 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 48 u^{2} + 24 u = 0

Solve with the quadratic formula

- 48 u^{2} + 24 u = 0

Quadratic Equation Formula:

For

a = - 48, b = 24, c = 0

u_{1, 2} = \frac{- 24 \pm 2 4 ^{2} - 4 ( - 48 ) \cdot 0}{2 ( - 48 )}

u_{1, 2} = \frac{- 24 \pm 2 4 ^{2} - 4 ( - 48 ) \cdot 0}{2 ( - 48 )}

2 4^{2} - 4 (- 48) \cdot 0 = 24

2 4^{2} - 4 (- 48) \cdot 0

Apply rule

- (- a) = a

= 2 4^{2} + 4 \cdot 48 \cdot 0

Apply rule

0 \cdot a = 0

= 2 4^{2} + 0

2 4^{2} + 0 = 2 4^{2}

= 2 4^{2}

Apply radical rule:

assuming

a \geq 0

= 24

u_{1, 2} = \frac{- 24 \pm 24}{2 ( - 48 )}

Separate the solutions

u_{1} = \frac{- 24 + 24}{2 ( - 48 )}, u_{2} = \frac{- 24 - 24}{2 ( - 48 )}

u = \frac{- 24 + 24}{2 ( - 48 )} : 0

\frac{- 24 + 24}{2 ( - 48 )}

Remove parentheses:

(- a) = - a

= \frac{- 24 + 24}{- 2 \cdot 48}

Add/Subtract the numbers:

- 24 + 24 = 0

= \frac{0}{- 2 \cdot 48}

Multiply the numbers:

2 \cdot 48 = 96

= \frac{0}{- 96}

Apply the fraction rule:

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

= - \frac{0}{96}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

u = \frac{- 24 - 24}{2 ( - 48 )} : \frac{1}{2}

\frac{- 24 - 24}{2 ( - 48 )}

Remove parentheses:

(- a) = - a

= \frac{- 24 - 24}{- 2 \cdot 48}

Subtract the numbers:

- 24 - 24 = - 48

= \frac{- 48}{- 2 \cdot 48}

Multiply the numbers:

2 \cdot 48 = 96

= \frac{- 48}{- 96}

Apply the fraction rule:

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

= \frac{48}{96}

Cancel the common factor:

48

= \frac{1}{2}

The solutions to the quadratic equation are:

u = 0, u = \frac{1}{2}

Substitute back

u = cos (x)

cos (x) = 0, cos (x) = \frac{1}{2}

cos (x) = 0, cos (x) = \frac{1}{2}

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

General solutions for

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

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}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

Combine all the solutions

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

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