Answer: The correct option is
(B) [tex]\sin C=\cos A.[/tex]
Step-by-step explanation: We are given to select the true statement about the triangle ABC shown in the figure.
From the figure, we note that
triangle ABC is right-angled at angle B, where hypotenuse, AC = 13, BC = 5 and AB = 12.
From the trigonometric ratios, we have
[tex]\sin A=\dfrac{\textup{perpendicular}}{\textup{hypotenuse}}=\dfrac{BC}{AC}=\dfrac{5}{13},\\\\\\\cos A=\dfrac{\textup{base}}{\textup{hypotenuse}}=\dfrac{AB}{AC}=\dfrac{12}{13},\\\\\\\cos B=\dfrac{\textup{base}}{\textup{hypotenuse}}=\dfrac{BC}{AC}=\dfrac{5}{13},\\\\\\\sin C=\dfrac{\textup{perpendicular}}{\textup{hypotenuse}}=\dfrac{AB}{AC}=\dfrac{12}{13},\\\\\\\sin A=\dfrac{\textup{perpendicular}}{\textup{hypotenuse}}=\dfrac{BC}{AC}=\dfrac{5}{13},\\\\\\\tan A=\dfrac{\textup{perpendicular}}{\textup{base}}=\dfrac{BC}{AB}=\dfrac{5}{12}.[/tex]
Therefore, we get
[tex]\sin C=\cos A.[/tex]
Thus, option (B) is CORRECT.
