find the value of x by using the trigonometry ratio.

Step-by-step solution:

The figure is shown above:

Notice that:

triangle ABD is a Right Triangle:

\sin{D}=\frac{AB}{AD}

Substitute the AD=130, angle D=74 degree into the Sine:

\sin{(\textcolor{secondary}{74^{\circ}})}=\frac{{AB}}{\textcolor{primary}{130}}

cross multiply we get,

AB=\sin{(74^{\circ})}\times130=>AB\approx125

Similarly, For cosine:

\cos{D}=\frac{BD}{AD}

Substitute the AD=130, angle D=74 degree into the Cosine:

\cos{(\textcolor{secondary}{74^{\circ}})}=\frac{{BD}}{\textcolor{primary}{130}}

cross multiply we get,

BD=\cos{(74^{\circ})}\times130=>BD\approx36

Notice that triangle ABC is a Right Triangle:

tangent formula (tan(x)):

\tan{C}=\frac{AB}{BC}

Substitute the AB= 125,BC= (36+x), angle C=42 degree into the Tangent:

\tan{(\textcolor{secondary}{42^{\circ}})}=\frac{{\textcolor{primary}{125}}}{\textcolor{tertiary}{36+x}}

After simplifying we get,

x\approx103

The value of x is approximately 103.

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