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Popular Trigonometry >

10sin(20t-pi/6)=8

Solution

10 sin (20 t - \frac{π}{6}) = 8

Solution

t = \frac{πn}{10} + \frac{π}{120} + \frac{0.92729 \dots}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{0.92729 \dots}{20}

Degrees

t = 4.15650 \dots^{\circ} + 1 8^{\circ} n, t = 7.84349 \dots^{\circ} + 1 8^{\circ} n

Solution steps

10 sin (20 t - \frac{π}{6}) = 8

Divide both sides by

10

10 sin (20 t - \frac{π}{6}) = 8

Divide both sides by

10

\frac{10 sin ( 20 t - \frac{π}{6} )}{10} = \frac{8}{10}

Simplify

sin (20 t - \frac{π}{6}) = \frac{4}{5}

sin (20 t - \frac{π}{6}) = \frac{4}{5}

Apply trig inverse properties

sin (20 t - \frac{π}{6}) = \frac{4}{5}

General solutions for

sin (20 t - \frac{π}{6}) = \frac{4}{5}

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

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn, 20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn, 20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

Solve

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn : t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn

Move

\frac{π}{6}

to the right side

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn

Add

\frac{π}{6}

to both sides

20 t - \frac{π}{6} + \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

Simplify

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

Divide both sides by

20

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

Divide both sides by

20

\frac{20 t}{20} = \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

Simplify

\frac{20 t}{20} = \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

Simplify

\frac{20 t}{20} : t

\frac{20 t}{20}

Divide the numbers:

\frac{20}{20} = 1

= t

Simplify

\frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} : \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

\frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

Group like terms

= \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} + \frac{arcsin ( \frac{4}{5} )}{20}

\frac{2 πn}{20} = \frac{πn}{10}

\frac{2 πn}{20}

Cancel the common factor:

2

= \frac{πn}{10}

\frac{\frac{π}{6}}{20} = \frac{π}{120}

\frac{\frac{π}{6}}{20}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 20}

Multiply the numbers:

6 \cdot 20 = 120

= \frac{π}{120}

= \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

Solve

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn : t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

Move

\frac{π}{6}

to the right side

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

Add

\frac{π}{6}

to both sides

20 t - \frac{π}{6} + \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

Simplify

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

Divide both sides by

20

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

Divide both sides by

20

\frac{20 t}{20} = \frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

Simplify

\frac{20 t}{20} = \frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

Simplify

\frac{20 t}{20} : t

\frac{20 t}{20}

Divide the numbers:

\frac{20}{20} = 1

= t

Simplify

\frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} : \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

\frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

Group like terms

= \frac{π}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} - \frac{arcsin ( \frac{4}{5} )}{20}

\frac{2 πn}{20} = \frac{πn}{10}

\frac{2 πn}{20}

Cancel the common factor:

2

= \frac{πn}{10}

\frac{\frac{π}{6}}{20} = \frac{π}{120}

\frac{\frac{π}{6}}{20}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 20}

Multiply the numbers:

6 \cdot 20 = 120

= \frac{π}{120}

= \frac{π}{20} + \frac{πn}{10} + \frac{π}{120} - \frac{arcsin ( \frac{4}{5} )}{20}

Group like terms

= \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

Show solutions in decimal form

t = \frac{πn}{10} + \frac{π}{120} + \frac{0.92729 \dots}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{0.92729 \dots}{20}

Graph

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