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Popular Trigonometry >

1.47=((sin((x+60)/2))/(sin(30)))

Solution

1.47 = (\frac{sin ( \frac{x + 60}{2} )}{sin ( 3 0 ^{\circ} )})

Solution

x = 2 \cdot 0.82566 \dots + 72 0^{\circ} n - 60, x = 36 0^{\circ} - 2 \cdot 0.82566 \dots + 72 0^{\circ} n - 60

Radians

x = 2 \cdot 0.82566 \dots - 60 + 4 πn, x = 2 π - 2 \cdot 0.82566 \dots - 60 + 4 πn

Solution steps

1.47 = (\frac{sin ( \frac{x + 60}{2} )}{sin ( 3 0 ^{\circ} )})

Switch sides

\frac{sin ( \frac{x + 60}{2} )}{sin ( 3 0 ^{\circ} )} = 1.47

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{1}{2}

\frac{sin ( \frac{x + 60}{2} )}{\frac{1}{2}} = 1.47

Multiply both sides by

\frac{1}{2}

\frac{sin ( \frac{x + 60}{2} )}{\frac{1}{2}} = 1.47

Multiply both sides by

\frac{1}{2}

\frac{sin ( \frac{x + 60}{2} ) \frac{1}{2}}{\frac{1}{2}} = 1.47 \cdot \frac{1}{2}

Simplify

\frac{sin ( \frac{x + 60}{2} ) \frac{1}{2}}{\frac{1}{2}} = 1.47 \cdot \frac{1}{2}

Simplify

\frac{sin ( \frac{x + 60}{2} ) \frac{1}{2}}{\frac{1}{2}} : sin (\frac{x + 60}{2})

\frac{sin ( \frac{x + 60}{2} ) \frac{1}{2}}{\frac{1}{2}}

Apply the fraction rule:

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

= \frac{sin ( \frac{x + 60}{2} ) \frac{1}{2} \cdot 2}{1}

Apply the fraction rule:

\frac{a}{1} = a

= sin (\frac{x + 60}{2}) \frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2} sin (\frac{x + 60}{2})

Cancel the common factor:

2

= sin (\frac{x + 60}{2}) \cdot 1

Multiply:

sin (\frac{x + 60}{2}) \cdot 1 = sin (\frac{x + 60}{2})

= sin (\frac{x + 60}{2})

Simplify

1.47 \cdot \frac{1}{2} : 0.735

1.47 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 1.47}{2}

Multiply the numbers:

1 \cdot 1.47 = 1.47

= \frac{1.47}{2}

Divide the numbers:

\frac{1.47}{2} = 0.735

= 0.735

sin (\frac{x + 60}{2}) = 0.735

sin (\frac{x + 60}{2}) = 0.735

sin (\frac{x + 60}{2}) = 0.735

Apply trig inverse properties

sin (\frac{x + 60}{2}) = 0.735

General solutions for

sin (\frac{x + 60}{2}) = 0.735

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

\frac{x + 60}{2} = arcsin (0.735) + 36 0^{\circ} n, \frac{x + 60}{2} = 18 0^{\circ} - arcsin (0.735) + 36 0^{\circ} n

\frac{x + 60}{2} = arcsin (0.735) + 36 0^{\circ} n, \frac{x + 60}{2} = 18 0^{\circ} - arcsin (0.735) + 36 0^{\circ} n

Solve

\frac{x + 60}{2} = arcsin (0.735) + 36 0^{\circ} n : x = 2 arcsin (0.735) + 72 0^{\circ} n - 60

\frac{x + 60}{2} = arcsin (0.735) + 36 0^{\circ} n

Multiply both sides by

2

\frac{x + 60}{2} = arcsin (0.735) + 36 0^{\circ} n

Multiply both sides by

2

\frac{2 ( x + 60 )}{2} = 2 arcsin (0.735) + 2 \cdot 36 0^{\circ} n

Simplify

x + 60 = 2 arcsin (0.735) + 72 0^{\circ} n

x + 60 = 2 arcsin (0.735) + 72 0^{\circ} n

Move

60

to the right side

x + 60 = 2 arcsin (0.735) + 72 0^{\circ} n

Subtract

60

from both sides

x + 60 - 60 = 2 arcsin (0.735) + 72 0^{\circ} n - 60

Simplify

x = 2 arcsin (0.735) + 72 0^{\circ} n - 60

x = 2 arcsin (0.735) + 72 0^{\circ} n - 60

Solve

\frac{x + 60}{2} = 18 0^{\circ} - arcsin (0.735) + 36 0^{\circ} n : x = 36 0^{\circ} - 2 arcsin (0.735) + 72 0^{\circ} n - 60

\frac{x + 60}{2} = 18 0^{\circ} - arcsin (0.735) + 36 0^{\circ} n

Multiply both sides by

2

\frac{x + 60}{2} = 18 0^{\circ} - arcsin (0.735) + 36 0^{\circ} n

Multiply both sides by

2

\frac{2 ( x + 60 )}{2} = 36 0^{\circ} - 2 arcsin (0.735) + 2 \cdot 36 0^{\circ} n

Simplify

x + 60 = 36 0^{\circ} - 2 arcsin (0.735) + 72 0^{\circ} n

x + 60 = 36 0^{\circ} - 2 arcsin (0.735) + 72 0^{\circ} n

Move

60

to the right side

x + 60 = 36 0^{\circ} - 2 arcsin (0.735) + 72 0^{\circ} n

Subtract

60

from both sides

x + 60 - 60 = 36 0^{\circ} - 2 arcsin (0.735) + 72 0^{\circ} n - 60

Simplify

x = 36 0^{\circ} - 2 arcsin (0.735) + 72 0^{\circ} n - 60

x = 36 0^{\circ} - 2 arcsin (0.735) + 72 0^{\circ} n - 60

x = 2 arcsin (0.735) + 72 0^{\circ} n - 60, x = 36 0^{\circ} - 2 arcsin (0.735) + 72 0^{\circ} n - 60

Show solutions in decimal form

x = 2 \cdot 0.82566 \dots + 72 0^{\circ} n - 60, x = 36 0^{\circ} - 2 \cdot 0.82566 \dots + 72 0^{\circ} n - 60

Graph

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