Answer:
Sin(B) = 15/17
Tan(B) = 15/8
Cos(D) = 15/17
Tan(D) = 8/15
Step-by-step explanation:
In order to determine the trigonometric ratios, we must first find the length of leg CD in right triangle BCD.
To do this, we can use the Pythagorean Theorem, which states that the square of the hypotenuse of a right triangle equals the sum of the squares of its legs. Therefore:
[tex]CD^2+BC^2=BD^2\\\\CD^2+8^2=17^2\\\\CD^2+64=289\\\\CD^2=289-64\\\\CD^2=225\\\\CD=\sqrt{225}\\\\CD=15[/tex]
So, leg CD measures 15 units.
Now that we have the lengths of all three sides of right triangle BCD, we can use the trigonometric ratios:
[tex]\boxed{\begin{array}{l}\underline{\sf Trigonometric\;ratios}\\\\\sf \sin(\theta)=\dfrac{O}{H}\qquad\cos(\theta)=\dfrac{A}{H}\qquad\tan(\theta)=\dfrac{O}{A}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{O is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{A is the side adjacent the angle.}\\\phantom{ww}\bullet\;\textsf{H is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]
For angle B, the side opposite the angle is CD, the adjacent side is BC, and the hypotenuse is BD. Therefore:
[tex]\sin (B)=\dfrac{CD}{BD}=\dfrac{15}{17}\\\\\\\tan (B)=\dfrac{CD}{BC}=\dfrac{15}{8}[/tex]
For angle D, the side opposite the angle is BC, the adjacent side is CD, and the hypotenuse is BD. Therefore:
[tex]\cos (D)=\dfrac{CD}{BD}=\dfrac{15}{17}\\\\\\\tan (D)=\dfrac{BC}{CD}=\dfrac{8}{15}[/tex]
