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Popular Trigonometry >

16sin^2(x)=16-8cos(x)

Solution

16 sin^{2} (x) = 16 - 8 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

16 sin^{2} (x) = 16 - 8 cos (x)

Subtract

16 - 8 cos (x)

from both sides

16 sin^{2} (x) - 16 + 8 cos (x) = 0

Rewrite using trig identities

- 16 + 16 sin^{2} (x) + 8 cos (x)

Use the Pythagorean identity:

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

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

= - 16 + 16 (1 - cos^{2} (x)) + 8 cos (x)

Simplify

- 16 + 16 (1 - cos^{2} (x)) + 8 cos (x) : 8 cos (x) - 16 cos^{2} (x)

- 16 + 16 (1 - cos^{2} (x)) + 8 cos (x)

Expand

16 (1 - cos^{2} (x)) : 16 - 16 cos^{2} (x)

16 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 16, b = 1, c = cos^{2} (x)

= 16 \cdot 1 - 16 cos^{2} (x)

Multiply the numbers:

16 \cdot 1 = 16

= 16 - 16 cos^{2} (x)

= - 16 + 16 - 16 cos^{2} (x) + 8 cos (x)

- 16 + 16 = 0

= 8 cos (x) - 16 cos^{2} (x)

= 8 cos (x) - 16 cos^{2} (x)

- 16 cos^{2} (x) + 8 cos (x) = 0

Solve by substitution

- 16 cos^{2} (x) + 8 cos (x) = 0

Let:

cos (x) = u

- 16 u^{2} + 8 u = 0

- 16 u^{2} + 8 u = 0 : u = 0, u = \frac{1}{2}

- 16 u^{2} + 8 u = 0

Solve with the quadratic formula

- 16 u^{2} + 8 u = 0

Quadratic Equation Formula:

For

a = - 16, b = 8, c = 0

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 16 ) \cdot 0}{2 ( - 16 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 16 ) \cdot 0}{2 ( - 16 )}

8^{2} - 4 (- 16) \cdot 0 = 8

8^{2} - 4 (- 16) \cdot 0

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 16 \cdot 0

Apply rule

0 \cdot a = 0

= 8^{2} + 0

8^{2} + 0 = 8^{2}

= 8^{2}

Apply radical rule:

n a^{n} = a,

assuming

a \geq 0

= 8

u_{1, 2} = \frac{- 8 \pm 8}{2 ( - 16 )}

Separate the solutions

u_{1} = \frac{- 8 + 8}{2 ( - 16 )}, u_{2} = \frac{- 8 - 8}{2 ( - 16 )}

u = \frac{- 8 + 8}{2 ( - 16 )} : 0

\frac{- 8 + 8}{2 ( - 16 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 8}{- 2 \cdot 16}

Add/Subtract the numbers:

- 8 + 8 = 0

= \frac{0}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{0}{- 32}

Apply the fraction rule:

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

= - \frac{0}{32}

Apply rule

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

= - 0

= 0

u = \frac{- 8 - 8}{2 ( - 16 )} : \frac{1}{2}

\frac{- 8 - 8}{2 ( - 16 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 8}{- 2 \cdot 16}

Subtract the numbers:

- 8 - 8 = - 16

= \frac{- 16}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{- 16}{- 32}

Apply the fraction rule:

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

= \frac{16}{32}

Cancel the common factor:

16

= \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

Graph

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