Answer:
The ratios are as follows:
[tex]\sin D=\dfrac{4}{5}[/tex]
[tex]\sin C=\dfrac{3}{5}[/tex]
[tex]\sin D\times \cos D=\dfrac{12}{25}[/tex]
[tex]\tan C\times \tan D=1[/tex]
[tex]\cos C\times \tan D=\dfrac{16}{15}[/tex]
Step-by-step explanation:
The trignometric ratio are defined as follows:
[tex]\sin \theta=\dfrac{opposite\ side}{hypotenuse}[/tex]
where opposite side is the side opposite to angle theta
and
[tex]\cos \theta=\dfrac{adjacent\ side}{hypotenuse}[/tex]
and adjacent side the side which is adjacent to angle θ and hypotenuse is the hypotenuse of the right angled triangle.
and
[tex]\tan \theta=\dfrac{opposite\ side}{adjacent\ side}[/tex]
Hence, from the right angled triangle that is provided to us we have:
[tex]\sin D=\dfrac{4}{5}[/tex]
[tex]\sin C=\dfrac{3}{5}[/tex]
and [tex]\cos D=\dfrac{3}{5}[/tex]
[tex]\cos C=\dfrac{4}{5}[/tex]
[tex]\tan C=\dfrac{3}{4}[/tex]
and [tex]\tan D=\dfrac{4}{3}[/tex]
Hence,
[tex]\sin D\times \cos D=\dfrac{4}{5}\times \dfrac{3}{5}\\\\\\\sin D\times \cos D=\dfrac{12}{25}[/tex]
[tex]\tan C\times \tan D=\dfrac{3}{4}\times \dfrac{4}{3}\\\\\\\tan C\times \tan D=1[/tex]
and
[tex]\cos C\times \tan D=\dfrac{4}{5}\times \dfrac{4}{3}\\\\\\\cos C\times \tan D=\dfrac{16}{15}[/tex]
