solvefor x,(17)/(sin(x))=(23)/(sin(75))

Solution

solve for

x, \frac{17}{sin ( x )} = \frac{23}{sin ( 7 5 ^{\circ} )}

x = 0.79511 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.79511 \dots + 36 0^{\circ} n

Solution steps

\frac{17}{sin ( x )} = \frac{23}{sin ( 7 5 ^{\circ} )}

sin (7 5^{\circ}) = \frac{6 + 2}{4}

\frac{17}{sin ( x )} = \frac{23}{\frac{6 + 2}{4}}

Cross multiply

\frac{17 ( 6 + 2 )}{4} = sin (x) \cdot 23

Switch sides

sin (x) \cdot 23 = \frac{17 ( 6 + 2 )}{4}

Divide both sides by

23

sin (x) = \frac{17 ( 6 + 2 )}{92}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{17 ( 6 + 2 )}{92}

Apply trig inverse properties

x = arcsin (\frac{17 ( 6 + 2 )}{92}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{17 ( 6 + 2 )}{92}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.79511 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.79511 \dots + 36 0^{\circ} n

3 - 2 cos^{2} (x) = 4 sin^{2} (x)

4 - 4 cos (2 x) - 3 sin (x) = 0

2 sin^{2} (w) + sin (w) - 1 = 0

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arctan (x) = arccos (\frac{3}{4})