What is the general solution for 7=4sin(pi/2 (x-2))+5 ?

The general solution for 7=4sin(pi/2 (x-2))+5 is x=4n+7/3 ,x=4n+11/3

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7=4sin(pi/2 (x-2))+5

Solution

7 = 4 sin (\frac{π}{2} (x - 2)) + 5

Solution

x = 4 n + \frac{7}{3}, x = 4 n + \frac{11}{3}

Degrees

x = 133.69015 \dots^{\circ} + 229.18311 \dots^{\circ} n, x = 210.08452 \dots^{\circ} + 229.18311 \dots^{\circ} n

Solution steps

7 = 4 sin (\frac{π}{2} (x - 2)) + 5

Switch sides

4 sin (\frac{π}{2} (x - 2)) + 5 = 7

Move

5

to the right side

4 sin (\frac{π}{2} (x - 2)) + 5 = 7

Subtract

5

from both sides

4 sin (\frac{π}{2} (x - 2)) + 5 - 5 = 7 - 5

Simplify

4 sin (\frac{π}{2} (x - 2)) = 2

4 sin (\frac{π}{2} (x - 2)) = 2

Divide both sides by

4

4 sin (\frac{π}{2} (x - 2)) = 2

Divide both sides by

4

\frac{4 sin ( \frac{π}{2} ( x - 2 ) )}{4} = \frac{2}{4}

Simplify

sin (\frac{π}{2} (x - 2)) = \frac{1}{2}

sin (\frac{π}{2} (x - 2)) = \frac{1}{2}

General solutions for

sin (\frac{π}{2} (x - 2)) = \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{π}{2} (x - 2) = \frac{π}{6} + 2 πn, \frac{π}{2} (x - 2) = \frac{5 π}{6} + 2 πn

\frac{π}{2} (x - 2) = \frac{π}{6} + 2 πn, \frac{π}{2} (x - 2) = \frac{5 π}{6} + 2 πn

Solve

\frac{π}{2} (x - 2) = \frac{π}{6} + 2 πn : x = 4 n + \frac{7}{3}

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

2 \cdot \frac{π}{2} (x - 2) : π (x - 2)

2 \cdot \frac{π}{2} (x - 2)

Multiply fractions:

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

= \frac{2 π}{2} (x - 2)

Cancel the common factor:

2

= (x - 2) π

Simplify

2 \cdot \frac{π}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

π (x - 2) = \frac{π}{3} + 4 πn

π (x - 2) = \frac{π}{3} + 4 πn

π (x - 2) = \frac{π}{3} + 4 πn

Divide both sides by

π

π (x - 2) = \frac{π}{3} + 4 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π ( x - 2 )}{π} : x - 2

\frac{π ( x - 2 )}{π}

Cancel the common factor:

π

= x - 2

Simplify

\frac{\frac{π}{3}}{π} + \frac{4 πn}{π} : \frac{1}{3} + 4 n

\frac{\frac{π}{3}}{π} + \frac{4 πn}{π}

\frac{\frac{π}{3}}{π} = \frac{1}{3}

\frac{\frac{π}{3}}{π}

Apply the fraction rule:

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

= \frac{π}{3 π}

Cancel the common factor:

π

= \frac{1}{3}

\frac{4 πn}{π} = 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= \frac{1}{3} + 4 n

x - 2 = \frac{1}{3} + 4 n

x - 2 = \frac{1}{3} + 4 n

x - 2 = \frac{1}{3} + 4 n

Move

2

to the right side

x - 2 = \frac{1}{3} + 4 n

Add

2

to both sides

x - 2 + 2 = \frac{1}{3} + 4 n + 2

Simplify

x - 2 + 2 = \frac{1}{3} + 4 n + 2

Simplify

x - 2 + 2 : x

x - 2 + 2

Add similar elements:

- 2 + 2 = 0

= x

Simplify

\frac{1}{3} + 4 n + 2 : 4 n + \frac{7}{3}

\frac{1}{3} + 4 n + 2

Combine the fractions

2 + \frac{1}{3} : \frac{7}{3}

2 + \frac{1}{3}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

2 \cdot 3 + 1 = 7

2 \cdot 3 + 1

Multiply the numbers:

2 \cdot 3 = 6

= 6 + 1

Add the numbers:

6 + 1 = 7

= 7

= \frac{7}{3}

= 4 n + \frac{7}{3}

x = 4 n + \frac{7}{3}

x = 4 n + \frac{7}{3}

x = 4 n + \frac{7}{3}

Solve

\frac{π}{2} (x - 2) = \frac{5 π}{6} + 2 πn : x = 4 n + \frac{11}{3}

\frac{π}{2} (x - 2) = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{π}{2} (x - 2) = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

2 \cdot \frac{π}{2} (x - 2) = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{π}{2} (x - 2) = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{π}{2} (x - 2) : π (x - 2)

2 \cdot \frac{π}{2} (x - 2)

Multiply fractions:

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

= \frac{2 π}{2} (x - 2)

Cancel the common factor:

2

= (x - 2) π

Simplify

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

π (x - 2) = \frac{5 π}{3} + 4 πn

π (x - 2) = \frac{5 π}{3} + 4 πn

π (x - 2) = \frac{5 π}{3} + 4 πn

Divide both sides by

π

π (x - 2) = \frac{5 π}{3} + 4 πn

Divide both sides by

π

\frac{π ( x - 2 )}{π} = \frac{\frac{5 π}{3}}{π} + \frac{4 πn}{π}

Simplify

\frac{π ( x - 2 )}{π} = \frac{\frac{5 π}{3}}{π} + \frac{4 πn}{π}

Simplify

\frac{π ( x - 2 )}{π} : x - 2

\frac{π ( x - 2 )}{π}

Cancel the common factor:

π

= x - 2

Simplify

\frac{\frac{5 π}{3}}{π} + \frac{4 πn}{π} : \frac{5}{3} + 4 n

\frac{\frac{5 π}{3}}{π} + \frac{4 πn}{π}

\frac{\frac{5 π}{3}}{π} = \frac{5}{3}

\frac{\frac{5 π}{3}}{π}

Apply the fraction rule:

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

= \frac{5 π}{3 π}

Cancel the common factor:

π

= \frac{5}{3}

\frac{4 πn}{π} = 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= \frac{5}{3} + 4 n

x - 2 = \frac{5}{3} + 4 n

x - 2 = \frac{5}{3} + 4 n

x - 2 = \frac{5}{3} + 4 n

Move

2

to the right side

x - 2 = \frac{5}{3} + 4 n

Add

2

to both sides

x - 2 + 2 = \frac{5}{3} + 4 n + 2

Simplify

x - 2 + 2 = \frac{5}{3} + 4 n + 2

Simplify

x - 2 + 2 : x

x - 2 + 2

Add similar elements:

- 2 + 2 = 0

= x

Simplify

\frac{5}{3} + 4 n + 2 : 4 n + \frac{11}{3}

\frac{5}{3} + 4 n + 2

Combine the fractions

2 + \frac{5}{3} : \frac{11}{3}

2 + \frac{5}{3}

Convert element to fraction:

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

= \frac{2 \cdot 3}{3} + \frac{5}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{2 \cdot 3 + 5}{3}

2 \cdot 3 + 5 = 11

2 \cdot 3 + 5

Multiply the numbers:

2 \cdot 3 = 6

= 6 + 5

Add the numbers:

6 + 5 = 11

= 11

= \frac{11}{3}

= 4 n + \frac{11}{3}

x = 4 n + \frac{11}{3}

x = 4 n + \frac{11}{3}

x = 4 n + \frac{11}{3}

x = 4 n + \frac{7}{3}, x = 4 n + \frac{11}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for 7=4sin(pi/2 (x-2))+5 ?
The general solution for 7=4sin(pi/2 (x-2))+5 is x=4n+7/3 ,x=4n+11/3