Question

Use the Pythagorean Theorem to find the missing leg of the right triangle and the. Identify the trig ratios

Answer 1

The Pythagorean theorem states:

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

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

In triangle EDF, DE = 29 is the hypotenuse, and FE = 20 and DF are the legs. Substituting this information into the formula and solving for DF, we get:

$\begin{gathered} DE^2=FE^2+DF^2 \\ 29^2=20^2+DF^2 \\ 841=400+DF^2 \\ 841-400=DF^2 \\ \sqrt[]{441}=DF \\ 21=DF \end{gathered}$

Sine formula

$\sin (angle)=\frac{\text{opposite side}}{hypotenuse}$

Considering angle D, the opposite side is FE, then:

$\begin{gathered} \sin D=\frac{FE}{DE} \\ \sin D=\frac{20}{29} \end{gathered}$

Considering angle E, the opposite side is DF, then:

$\begin{gathered} \sin E=\frac{DF}{DE} \\ \sin E=\frac{21}{29} \end{gathered}$

Cosine formula

$\cos (angle)=\frac{\text{adjacent side}}{hypotenuse}$

Considering angle D, the adjacent side is DF, then:

$\begin{gathered} \cos D=\frac{DF}{DE} \\ \cos D=\frac{21}{29} \end{gathered}$

Considering angle E, the adjacent side is FE, then:

$\begin{gathered} \cos E=\frac{FE}{DE} \\ \cos E=\frac{20}{29} \end{gathered}$

Tangent formula

$\tan (angle)=\frac{\text{opposite side}}{adjacent\text{ side}}$

Considering angle D, the opposite side is FE and the adjacent side is DF, then:

$\begin{gathered} \tan D=\frac{FE}{DF} \\ \tan D=\frac{20}{21} \end{gathered}$

Considering angle E, the opposite side is DF and the adjacent side is FE, then:

$\begin{gathered} \tan E=\frac{DF}{FE} \\ \tan E=\frac{21}{20} \end{gathered}$