Considering the right triangle ABC
Side a is opposite to ∠A and adjacent to ∠B
Side b is opposite to ∠B and adjacent to ∠A
To determine the length of "a" given that we know the length of b and the measure of ∠B, you have to apply the trigonometric ratio of the tangent which is defined as follows:
[tex]\tan \theta=\frac{opposite}{adjacent}[/tex]
The tangent of an angle "θ" is equal to the quotient between the opposite side of the angle and the adjacent side.
As mentioned before, considering ∠B, side b is opposite to this angle, and side a is adjacent to it.
Replace ∠B=70º and b=16 into the expression of the tangent:
[tex]\begin{gathered} \tan B=\frac{b}{a} \\ \tan 70=\frac{16}{a} \end{gathered}[/tex]
Multiply both sides by "a" to take the term from the denominator's place:
[tex]\begin{gathered} a\tan 70=a\cdot\frac{16}{a} \\ a\tan 70=16 \end{gathered}[/tex]
Divide both sides by the tangent of 70 to determine the length of a:
[tex]\begin{gathered} \frac{a\tan70}{\tan70}=\frac{16}{\tan 70} \\ a=\frac{16}{\tan 70} \\ a=5.82\approx5.8 \end{gathered}[/tex]
Side a is 5.8 units long. (first option)
