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Popular Trigonometry >

100+60sin((7pi)/3 x)=110,0<= x<= 1

Solution

100 + 60 sin (\frac{7 π}{3} x) = 110, 0 \leq x \leq 1

Solution

x = \frac{3 \cdot 0.16744 \dots}{7 π}, x = \frac{3 π - 3 \cdot 0.16744 \dots}{7 π}, x = \frac{3 \cdot 0.16744 \dots + 6 π}{7 π}

Degrees

x = 1.30880 \dots^{\circ}, x = 23.24652 \dots^{\circ}, x = 50.41947 \dots^{\circ}

Solution steps

100 + 60 sin (\frac{7 π}{3} x) = 110, 0 \leq x \leq 1

Move

100

to the right side

100 + 60 sin (\frac{7 π}{3} x) = 110

Subtract

100

from both sides

100 + 60 sin (\frac{7 π}{3} x) - 100 = 110 - 100

Simplify

60 sin (\frac{7 π}{3} x) = 10

60 sin (\frac{7 π}{3} x) = 10

Divide both sides by

60

60 sin (\frac{7 π}{3} x) = 10

Divide both sides by

60

\frac{60 sin ( \frac{7 π}{3} x )}{60} = \frac{10}{60}

Simplify

sin (\frac{7 π}{3} x) = \frac{1}{6}

sin (\frac{7 π}{3} x) = \frac{1}{6}

Apply trig inverse properties

sin (\frac{7 π}{3} x) = \frac{1}{6}

General solutions for

sin (\frac{7 π}{3} x) = \frac{1}{6}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

\frac{7 π}{3} x = arcsin (\frac{1}{6}) + 2 πn, \frac{7 π}{3} x = π - arcsin (\frac{1}{6}) + 2 πn

\frac{7 π}{3} x = arcsin (\frac{1}{6}) + 2 πn, \frac{7 π}{3} x = π - arcsin (\frac{1}{6}) + 2 πn

Solve

\frac{7 π}{3} x = arcsin (\frac{1}{6}) + 2 πn : x = \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

\frac{7 π}{3} x = arcsin (\frac{1}{6}) + 2 πn

Multiply both sides by

3

\frac{7 π}{3} x = arcsin (\frac{1}{6}) + 2 πn

Multiply both sides by

3

3 \cdot \frac{7 π}{3} x = 3 arcsin (\frac{1}{6}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{7 π}{3} x = 3 arcsin (\frac{1}{6}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{7 π}{3} x : 7 π x

3 \cdot \frac{7 π}{3} x

Multiply fractions:

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

= \frac{7 \cdot 3 π}{3} x

Cancel the common factor:

3

= x \cdot 7 π

Simplify

3 arcsin (\frac{1}{6}) + 3 \cdot 2 πn : 3 arcsin (\frac{1}{6}) + 6 πn

3 arcsin (\frac{1}{6}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 arcsin (\frac{1}{6}) + 6 πn

7 π x = 3 arcsin (\frac{1}{6}) + 6 πn

7 π x = 3 arcsin (\frac{1}{6}) + 6 πn

7 π x = 3 arcsin (\frac{1}{6}) + 6 πn

Divide both sides by

7 π

7 π x = 3 arcsin (\frac{1}{6}) + 6 πn

Divide both sides by

7 π

\frac{7 π x}{7 π} = \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 πn}{7 π}

Simplify

x = \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

x = \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

Solve

\frac{7 π}{3} x = π - arcsin (\frac{1}{6}) + 2 πn : x = \frac{3}{7} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

\frac{7 π}{3} x = π - arcsin (\frac{1}{6}) + 2 πn

Multiply both sides by

3

\frac{7 π}{3} x = π - arcsin (\frac{1}{6}) + 2 πn

Multiply both sides by

3

3 \cdot \frac{7 π}{3} x = 3 π - 3 arcsin (\frac{1}{6}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{7 π}{3} x = 3 π - 3 arcsin (\frac{1}{6}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{7 π}{3} x : 7 π x

3 \cdot \frac{7 π}{3} x

Multiply fractions:

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

= \frac{7 \cdot 3 π}{3} x

Cancel the common factor:

3

= x \cdot 7 π

Simplify

3 π - 3 arcsin (\frac{1}{6}) + 3 \cdot 2 πn : 3 π - 3 arcsin (\frac{1}{6}) + 6 πn

3 π - 3 arcsin (\frac{1}{6}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 π - 3 arcsin (\frac{1}{6}) + 6 πn

7 π x = 3 π - 3 arcsin (\frac{1}{6}) + 6 πn

7 π x = 3 π - 3 arcsin (\frac{1}{6}) + 6 πn

7 π x = 3 π - 3 arcsin (\frac{1}{6}) + 6 πn

Divide both sides by

7 π

7 π x = 3 π - 3 arcsin (\frac{1}{6}) + 6 πn

Divide both sides by

7 π

\frac{7 π x}{7 π} = \frac{3 π}{7 π} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 πn}{7 π}

Simplify

\frac{7 π x}{7 π} = \frac{3 π}{7 π} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 πn}{7 π}

Simplify

\frac{7 π x}{7 π} : x

\frac{7 π x}{7 π}

Divide the numbers:

\frac{7}{7} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{3 π}{7 π} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 πn}{7 π} : \frac{3}{7} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

\frac{3 π}{7 π} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 πn}{7 π}

Cancel

\frac{3 π}{7 π} : \frac{3}{7}

\frac{3 π}{7 π}

Cancel the common factor:

π

= \frac{3}{7}

= \frac{3}{7} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 πn}{7 π}

Cancel

\frac{6 πn}{7 π} : \frac{6 n}{7}

\frac{6 πn}{7 π}

Cancel the common factor:

π

= \frac{6 n}{7}

= \frac{3}{7} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

x = \frac{3}{7} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

x = \frac{3}{7} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

x = \frac{3}{7} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

x = \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}, x = \frac{3}{7} - \frac{3 arcsin ( \frac{1}{6} )}{7 π} + \frac{6 n}{7}

Solutions for the range

0 \leq x \leq 1

x = \frac{3 arcsin ( \frac{1}{6} )}{7 π}, x = \frac{3 π - 3 arcsin ( \frac{1}{6} )}{7 π}, x = \frac{3 arcsin ( \frac{1}{6} ) + 6 π}{7 π}

Show solutions in decimal form

x = \frac{3 \cdot 0.16744 \dots}{7 π}, x = \frac{3 π - 3 \cdot 0.16744 \dots}{7 π}, x = \frac{3 \cdot 0.16744 \dots + 6 π}{7 π}

Graph

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