What is the general solution for 24=32+8-2*sqrt(32*8)*cos(θ) ?

The general solution for 24=32+8-2*sqrt(32*8)*cos(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin

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Popular Trigonometry >

24=32+8-2sqrt(328)*cos(θ)

Solution

24 = 32 + 8 - 2 \cdot 32 \cdot 8 \cdot cos (θ)

Solution

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

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

24 = 32 + 8 - 2 32 \cdot 8 cos (θ)

Switch sides

32 + 8 - 2 32 \cdot 8 cos (θ) = 24

2 32 \cdot 8 cos (θ) = 32 cos (θ)

2 32 \cdot 8 cos (θ)

32 \cdot 8 = 16

32 \cdot 8

Multiply the numbers:

32 \cdot 8 = 256

= 256

Factor the number:

256 = 1 6^{2}

= 1 6^{2}

Apply radical rule:

n a^{n} = a

1 6^{2} = 16

= 16

= 2 \cdot 16 cos (θ)

Multiply the numbers:

2 \cdot 16 = 32

= 32 cos (θ)

32 + 8 - 32 cos (θ) = 24

Add the numbers:

32 + 8 = 40

- 32 cos (θ) + 40 = 24

Move

40

to the right side

- 32 cos (θ) + 40 = 24

Subtract

40

from both sides

- 32 cos (θ) + 40 - 40 = 24 - 40

Simplify

- 32 cos (θ) = - 16

- 32 cos (θ) = - 16

Divide both sides by

- 32

- 32 cos (θ) = - 16

Divide both sides by

- 32

\frac{- 32 cos ( θ )}{- 32} = \frac{- 16}{- 32}

Simplify

\frac{- 32 cos ( θ )}{- 32} = \frac{- 16}{- 32}

Simplify

\frac{- 32 cos ( θ )}{- 32} : cos (θ)

\frac{- 32 cos ( θ )}{- 32}

Apply the fraction rule:

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

= \frac{32 cos ( θ )}{32}

Divide the numbers:

\frac{32}{32} = 1

= cos (θ)

Simplify

\frac{- 16}{- 32} : \frac{1}{2}

\frac{- 16}{- 32}

Apply the fraction rule:

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

= \frac{16}{32}

Cancel the common factor:

16

= \frac{1}{2}

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

General solutions for

cos (θ) = \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}

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

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 24=32+8-2*sqrt(32*8)*cos(θ) ?
The general solution for 24=32+8-2*sqrt(32*8)*cos(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin