What is the general solution for 2sin(7/4 x)=-sqrt(3) ?

The general solution for 2sin(7/4 x)=-sqrt(3) is x=(16pi}{21}+(8pin)/7 ,x=(20pi)/(21)+\frac{8pin)/7

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2sin(7/4 x)=-sqrt(3)

Solution

2 sin (\frac{7}{4} x) = - 3

Solution

x = \frac{16 π}{21} + \frac{8 πn}{7}, x = \frac{20 π}{21} + \frac{8 πn}{7}

Degrees

x = 137.14285 \dots^{\circ} + 205.71428 \dots^{\circ} n, x = 171.42857 \dots^{\circ} + 205.71428 \dots^{\circ} n

Solution steps

2 sin (\frac{7}{4} x) = - 3

Divide both sides by

2

2 sin (\frac{7}{4} x) = - 3

Divide both sides by

2

\frac{2 sin ( \frac{7}{4} x )}{2} = \frac{- 3}{2}

Simplify

sin (\frac{7}{4} x) = - \frac{3}{2}

sin (\frac{7}{4} x) = - \frac{3}{2}

General solutions for

sin (\frac{7}{4} x) = - \frac{3}{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{7}{4} x = \frac{4 π}{3} + 2 πn, \frac{7}{4} x = \frac{5 π}{3} + 2 πn

\frac{7}{4} x = \frac{4 π}{3} + 2 πn, \frac{7}{4} x = \frac{5 π}{3} + 2 πn

Solve

\frac{7}{4} x = \frac{4 π}{3} + 2 πn : x = \frac{16 π}{21} + \frac{8 πn}{7}

\frac{7}{4} x = \frac{4 π}{3} + 2 πn

Multiply both sides by

4

\frac{7}{4} x = \frac{4 π}{3} + 2 πn

Multiply both sides by

4

4 \cdot \frac{7}{4} x = 4 \cdot \frac{4 π}{3} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{7}{4} x = 4 \cdot \frac{4 π}{3} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{7}{4} x : 7 x

4 \cdot \frac{7}{4} x

Multiply fractions:

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

= \frac{7 \cdot 4}{4} x

Cancel the common factor:

4

= x \cdot 7

Simplify

4 \cdot \frac{4 π}{3} + 4 \cdot 2 πn : \frac{16 π}{3} + 8 πn

4 \cdot \frac{4 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{4 π}{3} = \frac{16 π}{3}

4 \cdot \frac{4 π}{3}

Multiply fractions:

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

= \frac{4 π 4}{3}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{16 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= \frac{16 π}{3} + 8 πn

7 x = \frac{16 π}{3} + 8 πn

7 x = \frac{16 π}{3} + 8 πn

7 x = \frac{16 π}{3} + 8 πn

Divide both sides by

7

7 x = \frac{16 π}{3} + 8 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{\frac{16 π}{3}}{7} + \frac{8 πn}{7}

Simplify

\frac{7 x}{7} = \frac{\frac{16 π}{3}}{7} + \frac{8 πn}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{16 π}{3}}{7} + \frac{8 πn}{7} : \frac{16 π}{21} + \frac{8 πn}{7}

\frac{\frac{16 π}{3}}{7} + \frac{8 πn}{7}

\frac{\frac{16 π}{3}}{7} = \frac{16 π}{21}

\frac{\frac{16 π}{3}}{7}

Apply the fraction rule:

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

= \frac{16 π}{3 \cdot 7}

Multiply the numbers:

3 \cdot 7 = 21

= \frac{16 π}{21}

= \frac{16 π}{21} + \frac{8 πn}{7}

x = \frac{16 π}{21} + \frac{8 πn}{7}

x = \frac{16 π}{21} + \frac{8 πn}{7}

x = \frac{16 π}{21} + \frac{8 πn}{7}

Solve

\frac{7}{4} x = \frac{5 π}{3} + 2 πn : x = \frac{20 π}{21} + \frac{8 πn}{7}

\frac{7}{4} x = \frac{5 π}{3} + 2 πn

Multiply both sides by

4

\frac{7}{4} x = \frac{5 π}{3} + 2 πn

Multiply both sides by

4

4 \cdot \frac{7}{4} x = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{7}{4} x = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{7}{4} x : 7 x

4 \cdot \frac{7}{4} x

Multiply fractions:

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

= \frac{7 \cdot 4}{4} x

Cancel the common factor:

4

= x \cdot 7

Simplify

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn : \frac{20 π}{3} + 8 πn

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{3} = \frac{20 π}{3}

4 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 4}{3}

Multiply the numbers:

5 \cdot 4 = 20

= \frac{20 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= \frac{20 π}{3} + 8 πn

7 x = \frac{20 π}{3} + 8 πn

7 x = \frac{20 π}{3} + 8 πn

7 x = \frac{20 π}{3} + 8 πn

Divide both sides by

7

7 x = \frac{20 π}{3} + 8 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{\frac{20 π}{3}}{7} + \frac{8 πn}{7}

Simplify

\frac{7 x}{7} = \frac{\frac{20 π}{3}}{7} + \frac{8 πn}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{20 π}{3}}{7} + \frac{8 πn}{7} : \frac{20 π}{21} + \frac{8 πn}{7}

\frac{\frac{20 π}{3}}{7} + \frac{8 πn}{7}

\frac{\frac{20 π}{3}}{7} = \frac{20 π}{21}

\frac{\frac{20 π}{3}}{7}

Apply the fraction rule:

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

= \frac{20 π}{3 \cdot 7}

Multiply the numbers:

3 \cdot 7 = 21

= \frac{20 π}{21}

= \frac{20 π}{21} + \frac{8 πn}{7}

x = \frac{20 π}{21} + \frac{8 πn}{7}

x = \frac{20 π}{21} + \frac{8 πn}{7}

x = \frac{20 π}{21} + \frac{8 πn}{7}

x = \frac{16 π}{21} + \frac{8 πn}{7}, x = \frac{20 π}{21} + \frac{8 πn}{7}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(7/4 x)=-sqrt(3) ?
The general solution for 2sin(7/4 x)=-sqrt(3) is x=(16pi}{21}+(8pin)/7 ,x=(20pi)/(21)+\frac{8pin)/7