What is the general solution for (150)/(sin(x))=(140.78)/(sin(115)) ?

The general solution for (150)/(sin(x))=(140.78)/(sin(115)) is x=1.30798…+360n,x=180-1.30798…+360n

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(150)/(sin(x))=(140.78)/(sin(115))

\frac{150}{sin ( x )} = \frac{140.78}{sin ( 11 5 ^{\circ} )}

x = 1.30798 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 1.30798 \dots + 36 0^{\circ} n

Radians

x = 1.30798 \dots + 2 πn, x = π - 1.30798 \dots + 2 πn

Solution steps

\frac{150}{sin ( x )} = \frac{140.78}{sin ( 11 5 ^{\circ} )}

Cross multiply

\frac{150}{sin ( x )} = \frac{140.78}{sin ( 11 5 ^{\circ} )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

150 sin (11 5^{\circ}) = sin (x) \cdot 140.78

150 sin (11 5^{\circ}) = sin (x) \cdot 140.78

Switch sides

sin (x) \cdot 140.78 = 150 sin (11 5^{\circ})

Divide both sides by

140.78

sin (x) \cdot 140.78 = 150 sin (11 5^{\circ})

Divide both sides by

140.78

\frac{sin ( x ) \cdot 140.78}{140.78} = \frac{150 sin ( 11 5 ^{\circ} )}{140.78}

Simplify

sin (x) = \frac{150 sin ( 11 5 ^{\circ} )}{140.78}

sin (x) = \frac{150 sin ( 11 5 ^{\circ} )}{140.78}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{150}{sin ( x )}

and compare to zero

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{150 sin ( 11 5 ^{\circ} )}{140.78}

Apply trig inverse properties

sin (x) = \frac{150 sin ( 11 5 ^{\circ} )}{140.78}

General solutions for

sin (x) = \frac{150 sin ( 11 5 ^{\circ} )}{140.78}

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

x = arcsin (\frac{150 sin ( 11 5 ^{\circ} )}{140.78}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{150 sin ( 11 5 ^{\circ} )}{140.78}) + 36 0^{\circ} n

x = arcsin (\frac{150 sin ( 11 5 ^{\circ} )}{140.78}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{150 sin ( 11 5 ^{\circ} )}{140.78}) + 36 0^{\circ} n

Show solutions in decimal form

x = 1.30798 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 1.30798 \dots + 36 0^{\circ} n

tan^{2} (x) + cot^{2} (x) = 2

tan^{2} (x) + cot^{2} (x) = 1

4 sin (θ) = - 4 cos (θ)

- 3 sin (a) - 2 = sin (a) - 2

tan (A) = 1

What is the general solution for (150)/(sin(x))=(140.78)/(sin(115)) ?
The general solution for (150)/(sin(x))=(140.78)/(sin(115)) is x=1.30798…+360n,x=180-1.30798…+360n