What is the general solution for 100cos(30)-200cos(θ)=0 ?

The general solution for 100cos(30)-200cos(θ)=0 is θ=1.12296…+360n,θ=360-1.12296…+360n

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100cos(30)-200cos(θ)=0

Solution

100 cos (3 0^{\circ}) - 200 cos (θ) = 0

Solution

θ = 1.12296 \dots + 36 0^{\circ} n, θ = 36 0^{\circ} - 1.12296 \dots + 36 0^{\circ} n

Radians

θ = 1.12296 \dots + 2 πn, θ = 2 π - 1.12296 \dots + 2 πn

Solution steps

100 cos (3 0^{\circ}) - 200 cos (θ) = 0

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2}

100 \cdot \frac{3}{2} - 200 cos (θ) = 0

100 \cdot \frac{3}{2} = 503

100 \cdot \frac{3}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 100}{2}

Divide the numbers:

\frac{100}{2} = 50

= 503

503 - 200 cos (θ) = 0

Move

503

to the right side

503 - 200 cos (θ) = 0

Subtract

503

from both sides

503 - 200 cos (θ) - 503 = 0 - 503

Simplify

- 200 cos (θ) = - 503

- 200 cos (θ) = - 503

Divide both sides by

- 200

- 200 cos (θ) = - 503

Divide both sides by

- 200

\frac{- 200 cos ( θ )}{- 200} = \frac{- 50 3}{- 200}

Simplify

\frac{- 200 cos ( θ )}{- 200} = \frac{- 50 3}{- 200}

Simplify

\frac{- 200 cos ( θ )}{- 200} : cos (θ)

\frac{- 200 cos ( θ )}{- 200}

Apply the fraction rule:

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

= \frac{200 cos ( θ )}{200}

Divide the numbers:

\frac{200}{200} = 1

= cos (θ)

Simplify

\frac{- 50 3}{- 200} : \frac{3}{4}

\frac{- 50 3}{- 200}

Apply the fraction rule:

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

= \frac{50 3}{200}

Cancel the common factor:

50

= \frac{3}{4}

cos (θ) = \frac{3}{4}

cos (θ) = \frac{3}{4}

cos (θ) = \frac{3}{4}

Apply trig inverse properties

cos (θ) = \frac{3}{4}

General solutions for

cos (θ) = \frac{3}{4}

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

θ = arccos (\frac{3}{4}) + 36 0^{\circ} n, θ = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n

θ = arccos (\frac{3}{4}) + 36 0^{\circ} n, θ = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n

Show solutions in decimal form

θ = 1.12296 \dots + 36 0^{\circ} n, θ = 36 0^{\circ} - 1.12296 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for 100cos(30)-200cos(θ)=0 ?
The general solution for 100cos(30)-200cos(θ)=0 is θ=1.12296…+360n,θ=360-1.12296…+360n