In ΔABC with AB = 5 cm, BC = 12 cm, AC = 13 cm and ∠B = 90°, which of the following is/are not correct?
(A) \(\tan C=\displaystyle\frac{12}{13}\)
(B) \(\mathrm{cosec}\, A=\displaystyle\frac{13}{12}\)
(C) \(\sin B=\displaystyle\frac{5}{13}\)
(D) \(\tan A=\displaystyle\frac{12}{15}\)
(E) \(\cos C=\displaystyle\frac{12}{13}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
In a right triangle, each ratio is taken from the angle’s point of view: the side facing the angle is the opposite side, the side next to it (not the hypotenuse) is the adjacent side, and the hypotenuse here is AC = 13 cm, facing the right angle at B.
For angle A: opposite side BC = 12, adjacent side AB = 5
For angle C: opposite side AB = 5, adjacent side BC = 12
(A) \(\tan C=\frac{12}{13}\) Not correct. For C, the opposite side is AB and the adjacent side is BC; the given value is \(\cos C\), not \(\tan C\).
\(\displaystyle \tan C=\frac{AB}{BC}=\frac{5}{12}\)
(B) \(\mathrm{cosec}\,A=\frac{13}{12}\) Correct. Cosec is hypotenuse over opposite side, and the side opposite A is BC.
\(\displaystyle \mathrm{cosec}\,A=\frac{AC}{BC}=\frac{13}{12}\)
(C) \(\sin B=\frac{5}{13}\) Not correct. B is the right angle, and the sine of 90° is 1.
(D) \(\tan A=\frac{12}{15}\) Not correct. The side adjacent to A is AB = 5 cm, so the denominator should be 5, not 15.
\(\displaystyle \tan A=\frac{BC}{AB}=\frac{12}{5}\)
(E) \(\cos C=\frac{12}{13}\) Correct. Cos is adjacent side over hypotenuse, and the side adjacent to C is BC.
\(\displaystyle \cos C=\frac{BC}{AC}=\frac{12}{13}\)