What is the general solution for 5+3cos(1/2 x)=7 ?

The general solution for 5+3cos(1/2 x)=7 is x=2*0.84106…+4pin,x=4pi-2*0.84106…+4pin

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5+3cos(1/2 x)=7

Solution

5 + 3 cos (\frac{1}{2} x) = 7

Solution

x = 2 \cdot 0.84106 \dots + 4 πn, x = 4 π - 2 \cdot 0.84106 \dots + 4 πn

Degrees

x = 96.37937 \dots^{\circ} + 72 0^{\circ} n, x = 623.62062 \dots^{\circ} + 72 0^{\circ} n

Solution steps

5 + 3 cos (\frac{1}{2} x) = 7

Move

5

to the right side

5 + 3 cos (\frac{1}{2} x) = 7

Subtract

5

from both sides

5 + 3 cos (\frac{1}{2} x) - 5 = 7 - 5

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

\frac{1}{2} x = arccos (\frac{2}{3}) + 2 πn, \frac{1}{2} x = 2 π - arccos (\frac{2}{3}) + 2 πn

\frac{1}{2} x = arccos (\frac{2}{3}) + 2 πn, \frac{1}{2} x = 2 π - arccos (\frac{2}{3}) + 2 πn

Solve

\frac{1}{2} x = arccos (\frac{2}{3}) + 2 πn : x = 2 arccos (\frac{2}{3}) + 4 πn

\frac{1}{2} x = arccos (\frac{2}{3}) + 2 πn

Multiply both sides by

2

\frac{1}{2} x = arccos (\frac{2}{3}) + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 arccos (\frac{2}{3}) + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x = 2 arccos (\frac{2}{3}) + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

2 arccos (\frac{2}{3}) + 2 \cdot 2 πn : 2 arccos (\frac{2}{3}) + 4 πn

2 arccos (\frac{2}{3}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arccos (\frac{2}{3}) + 4 πn

x = 2 arccos (\frac{2}{3}) + 4 πn

x = 2 arccos (\frac{2}{3}) + 4 πn

x = 2 arccos (\frac{2}{3}) + 4 πn

Solve

\frac{1}{2} x = 2 π - arccos (\frac{2}{3}) + 2 πn : x = 4 π - 2 arccos (\frac{2}{3}) + 4 πn

\frac{1}{2} x = 2 π - arccos (\frac{2}{3}) + 2 πn

Multiply both sides by

2

\frac{1}{2} x = 2 π - arccos (\frac{2}{3}) + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot 2 π - 2 arccos (\frac{2}{3}) + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot 2 π - 2 arccos (\frac{2}{3}) + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

2 \cdot 2 π - 2 arccos (\frac{2}{3}) + 2 \cdot 2 πn : 4 π - 2 arccos (\frac{2}{3}) + 4 πn

2 \cdot 2 π - 2 arccos (\frac{2}{3}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 π - 2 arccos (\frac{2}{3}) + 4 πn

x = 4 π - 2 arccos (\frac{2}{3}) + 4 πn

x = 4 π - 2 arccos (\frac{2}{3}) + 4 πn

x = 4 π - 2 arccos (\frac{2}{3}) + 4 πn

x = 2 arccos (\frac{2}{3}) + 4 πn, x = 4 π - 2 arccos (\frac{2}{3}) + 4 πn

Show solutions in decimal form

x = 2 \cdot 0.84106 \dots + 4 πn, x = 4 π - 2 \cdot 0.84106 \dots + 4 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5+3cos(1/2 x)=7 ?
The general solution for 5+3cos(1/2 x)=7 is x=2*0.84106…+4pin,x=4pi-2*0.84106…+4pin