What is the general solution for 1/2 =sqrt(3/2)*cos(x) ?

The general solution for 1/2 =sqrt(3/2)*cos(x) is x=1.15026…+2pin,x=2pi-1.15026…+2pin

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1/2 =sqrt(3/2)*cos(x)

Solution

\frac{1}{2} = \frac{3}{2} \cdot cos (x)

Solution

x = 1.15026 \dots + 2 πn, x = 2 π - 1.15026 \dots + 2 πn

Degrees

x = 65.90515 \dots^{\circ} + 36 0^{\circ} n, x = 294.09484 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{2} = \frac{3}{2} cos (x)

Switch sides

\frac{3}{2} cos (x) = \frac{1}{2}

Divide both sides by

\frac{3}{2}

\frac{3}{2} cos (x) = \frac{1}{2}

Divide both sides by

\frac{3}{2}

\frac{\frac{3}{2} cos ( x )}{\frac{3}{2}} = \frac{\frac{1}{2}}{\frac{3}{2}}

Simplify

\frac{\frac{3}{2} cos ( x )}{\frac{3}{2}} = \frac{\frac{1}{2}}{\frac{3}{2}}

Simplify

\frac{\frac{3}{2} cos ( x )}{\frac{3}{2}} : cos (x)

\frac{\frac{3}{2} cos ( x )}{\frac{3}{2}}

Cancel the common factor:

\frac{3}{2}

= cos (x)

Simplify

\frac{\frac{1}{2}}{\frac{3}{2}} : \frac{6}{6}

\frac{\frac{1}{2}}{\frac{3}{2}}

Apply the fraction rule:

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

= \frac{1}{2 \frac{3}{2}}

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

= \frac{1}{2 \cdot \frac{3}{2}}

Multiply

2 \cdot \frac{3}{2} : 6

2 \cdot \frac{3}{2}

Multiply fractions:

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

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

Apply radical rule:

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

= \frac{2 3}{2 ^{\frac{1}{2}}}

Apply exponent rule:

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

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

= 3 \cdot 2^{- \frac{1}{2} + 1}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 3 \cdot 2^{\frac{1}{2}}

Apply radical rule:

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

= 32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{1}{6}

Rationalize

\frac{1}{6} : \frac{6}{6}

\frac{1}{6}

Multiply by the conjugate

\frac{6}{6}

= \frac{1 \cdot 6}{6 6}

1 \cdot 6 = 6

66 = 6

66

Apply radical rule:

a a = a

66 = 6

= 6

= \frac{6}{6}

= \frac{6}{6}

cos (x) = \frac{6}{6}

cos (x) = \frac{6}{6}

cos (x) = \frac{6}{6}

Apply trig inverse properties

cos (x) = \frac{6}{6}

General solutions for

cos (x) = \frac{6}{6}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{6}{6}) + 2 πn, x = 2 π - arccos (\frac{6}{6}) + 2 πn

x = arccos (\frac{6}{6}) + 2 πn, x = 2 π - arccos (\frac{6}{6}) + 2 πn

Show solutions in decimal form

x = 1.15026 \dots + 2 πn, x = 2 π - 1.15026 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 1/2 =sqrt(3/2)*cos(x) ?
The general solution for 1/2 =sqrt(3/2)*cos(x) is x=1.15026…+2pin,x=2pi-1.15026…+2pin