Answer:
OPTION C: Sin C - Cos C = s - r
Step-by-step explanation:
ABC is a right angled triangle. ∠A = 90°, from the figure.
Therefore, BC = hypotenuse, say h
Now, we find the length of AB and AC.
We know that: [tex]$ \textbf{Sin A} = \frac{\textbf{opp}}{\textbf{hyp}} $[/tex]
and [tex]$ \textbf{Cos A} = \frac{\textbf{adj}}{\textbf{hyp}} $[/tex]
Given, Sin B = r and Cos B = s
⇒ [tex]$ Sin B = r = \frac{opp}{hyp} = \frac{AC}{BC} = \frac{AC}{h} $[/tex]
⇒ [tex]$ \textbf{AC} = \textbf{rh} $[/tex]
Hence, the length of the side AC = rh
Now, to compute the length of AB, we use Cos B.
[tex]$ Cos B = s = \frac{adj}{hyp} = \frac{AB}{BC} = \frac{AB}{h} $[/tex]
⇒ [tex]$ \textbf{AB} = \textbf{sh} $[/tex]
Hence, the length of the side AB = sh
Now, we are asked to compute Sin C - Cos C.
[tex]$ Sin C = \frac{opp}{hyp} $[/tex]
⇒ [tex]$ Sin C = \frac{AB}{BC} $[/tex]
[tex]$ = \frac{sh}{h} $[/tex]
= s
Sin C = s
[tex]$ Cos C = \frac{adj}{hyp} $[/tex]
[tex]$ \implies Cos C = \frac{AC}{BC} $[/tex]
⇒ Cos C = [tex]$ \frac{rh}{h} $[/tex]
Therefore, Cos C = r
So, Sin C - Cos C = s - r, which is OPTION C and is the right answer.
