Question

In triangle STU, the measure of angle U=90°, TS = 41, UT = 40, and SU = 9. What is the value ofthe tangent of angle S to the nearest hundredth?

Answer 1

so, we have a angle U=90 degrees it means , that this is a rigth triangle, we can use the Pithagoras theorem

is says

$a^2+b^2=c^2$

where, a and b are the legs of the triangle and c is the hypotenuse.

Also

TS=41

UT=40

SU=9

so, the longest side is the hypotenuse=TS=41

hypotenuse=41

Leg1=40

Leg2=9

$\begin{gathered} 9^2+40^2=41^2 \\ 81+1600=1681 \\ 1681=1681 \end{gathered}$

then, it is a rigth triangle, we can use trigonometric relations to find tan S

$\begin{gathered} \sin \emptyset=\frac{opposite\text{ side}}{\text{hypotenuse}} \\ \text{and } \\ \cos \emptyset\text{ =}\frac{adjacent\text{ side}}{\text{hypotenuse}} \\ \text{tan}\emptyset\text{ =}\frac{\sin \emptyset}{\cos \emptyset} \\ \text{tan}\emptyset=\frac{\frac{\text{opposite side}}{\text{hypotenuse}}}{\frac{\text{adjacent side}}{\text{hypotenuse}}} \\ \text{tan}\emptyset\text{ =}\frac{\text{opposite side}}{\text{adjacent side}} \end{gathered}$

Now, we have to identify opposite side and adjacen side.

for the angle S, the opposite side is TU=40

and the adjacen side is SU=9

replace the values

$\begin{gathered} \tan S=\frac{40}{9} \\ \tan S=4.44 \end{gathered}$

to the nearest hundredth

Tan S=4.44