Question

solving for side lengths of right triangles
determining trigonometric ratios
consider △dfe. what are the inputs or outputs of the following trigonometric ratios? express the ratios in simplest terms.
sin( ) = (\frac{?}{?})
cos( )
tan( )

Explanation:

Step1: Find the third - side length

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 25\) and \(a = 20\). Let the other side be \(b\). Then \(b=\sqrt{25^{2}-20^{2}}=\sqrt{(25 + 20)(25 - 20)}=\sqrt{45\times5}=\sqrt{225}=15\).

Step2: Recall trigonometric - ratio definitions

The sine of an angle in a right - triangle is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), the cosine is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), and the tangent is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For \(\triangle DFE\), if the vertical side is \(20\), the horizontal side is \(15\), and the hypotenuse is \(25\).
\(\sin(E)=\frac{20}{25}=\frac{4}{5}\), \(\cos(F)=\frac{20}{25}=\frac{4}{5}\), \(\tan(D)=\frac{15}{20}=\frac{3}{4}\).

Answer:

\(\sin(E)=\frac{4}{5}\), \(\cos(F)=\frac{4}{5}\), \(\tan(D)=\frac{3}{4}\)