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solvefor x,(sin(x))/5 =(sin(125))/(20)

Solution

solve for

x, \frac{sin ( x )}{5} = \frac{sin ( 12 5 ^{\circ} )}{20}

x = 0.20624 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.20624 \dots + 36 0^{\circ} n

Radians

x = 0.20624 \dots + 2 πn, x = π - 0.20624 \dots + 2 πn

Solution steps

\frac{sin ( x )}{5} = \frac{sin ( 12 5 ^{\circ} )}{20}

Multiply both sides by

10

\frac{sin ( x )}{5} = \frac{sin ( 12 5 ^{\circ} )}{20}

Multiply both sides by

10

\frac{sin ( x )}{5} \cdot 10 = \frac{sin ( 12 5 ^{\circ} )}{20} \cdot 10

Simplify

\frac{sin ( x )}{5} \cdot 10 = \frac{sin ( 12 5 ^{\circ} )}{20} \cdot 10

Simplify

\frac{sin ( x )}{5} \cdot 10 : 2 sin (x)

\frac{sin ( x )}{5} \cdot 10

Multiply fractions:

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

= \frac{sin ( x ) \cdot 10}{5}

Divide the numbers:

\frac{10}{5} = 2

= 2 sin (x)

Simplify

\frac{sin ( 12 5 ^{\circ} )}{20} \cdot 10 : \frac{sin ( 12 5 ^{\circ} )}{2}

\frac{sin ( 12 5 ^{\circ} )}{20} \cdot 10

Multiply fractions:

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

= \frac{sin ( 12 5 ^{\circ} ) \cdot 10}{20}

Cancel the common factor:

10

= \frac{sin ( 12 5 ^{\circ} )}{2}

2 sin (x) = \frac{sin ( 12 5 ^{\circ} )}{2}

2 sin (x) = \frac{sin ( 12 5 ^{\circ} )}{2}

2 sin (x) = \frac{sin ( 12 5 ^{\circ} )}{2}

Divide both sides by

2

2 sin (x) = \frac{sin ( 12 5 ^{\circ} )}{2}

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{\frac{s i n ( 12 5 ^{\circ} )}{2}}{2}

Simplify

\frac{2 sin ( x )}{2} = \frac{\frac{s i n ( 12 5 ^{\circ} )}{2}}{2}

Simplify

\frac{2 sin ( x )}{2} : sin (x)

\frac{2 sin ( x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= sin (x)

Simplify

\frac{\frac{s i n ( 12 5 ^{\circ} )}{2}}{2} : \frac{sin ( 12 5 ^{\circ} )}{4}

\frac{\frac{s i n ( 12 5 ^{\circ} )}{2}}{2}

Apply the fraction rule:

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

= \frac{sin ( 12 5 ^{\circ} )}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{sin ( 12 5 ^{\circ} )}{4}

sin (x) = \frac{sin ( 12 5 ^{\circ} )}{4}

sin (x) = \frac{sin ( 12 5 ^{\circ} )}{4}

sin (x) = \frac{sin ( 12 5 ^{\circ} )}{4}

Apply trig inverse properties

sin (x) = \frac{sin ( 12 5 ^{\circ} )}{4}

General solutions for

sin (x) = \frac{sin ( 12 5 ^{\circ} )}{4}

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

x = arcsin (\frac{sin ( 12 5 ^{\circ} )}{4}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{sin ( 12 5 ^{\circ} )}{4}) + 36 0^{\circ} n

x = arcsin (\frac{sin ( 12 5 ^{\circ} )}{4}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{sin ( 12 5 ^{\circ} )}{4}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.20624 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.20624 \dots + 36 0^{\circ} n

2 - 2 sin^{2} (x) = 0

sin (x) = \frac{2.5}{3}

tan (θ) + cot (θ) = 1

sin (θ) = - \frac{1}{2}, 0 \leq θ < 2 π

tan (x) = 3 + 2