Answer:
[tex]a=\sqrt{7}[/tex]
[tex]\text{tan}(A)=\frac{\sqrt{7}}{3}[/tex]
Step-by-step explanation:
Please find the attachment.
We have been given that ABC is a right triangle with sides of lengths a, b, and c and right angle at C.
To find the side length a, we will Pythagoras theorem, which states that the sum of squares of two legs of a right triangle is equal to the square of the hypotenuse of right triangle.
[tex]a^2+b^2=c^2[/tex]
Upon substituting our given values in Pythagoras theorem, we will get:
[tex]a^2+3^2=4^2[/tex]
[tex]a^2+9=16[/tex]
[tex]a^2+9-9=16-9[/tex]
[tex]a^2=7[/tex]
Take square root of both sides:
[tex]a=\sqrt{7}[/tex]
Therefore, the length of side 'a' is [tex]\sqrt{7}[/tex] units.
We know that tangent relates opposite side of a right triangle with adjacent side.
[tex]\text{tan}=\frac{\text{Opposite}}{\text{Adjacent}}[/tex]
We can see that 'a' is opposite side of angle A and 'b' is adjacent side.
[tex]\text{tan}(A)=\frac{a}{b}[/tex]
[tex]\text{tan}(A)=\frac{\sqrt{7}}{3}[/tex]
Therefore, the value of tan(A) is [tex]\frac{\sqrt{7}}{3}[/tex].
