question_answer

Assertion (A): In \[\Delta ABC,\] right angled at B, if \[\sin A=\frac{8}{17},\]then \[\cos A=\frac{15}{17}\]and \[\tan A=\frac{8}{15}\].

Reason (R): For acute angle \[\theta ,\] \[\cos \theta =\frac{Hypotenuse}{Base},\]and \[\tan \theta =\frac{Base}{Perpendicular}\].

A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)

B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A)

C) Assertion (A) is true but Reason (R) is false

D) Assertion (A) is false but Reason (R) is true

Correct Answer: C

Solution :

[c] Let us draw a \[\Delta ABC\] in which

\[\angle B={{90}^{o}}\] Then, \[\operatorname{sinA}=\frac{BC}{AC}=\frac{8}{17}\]

Let BC=8k and AC=17k where k is

Let BC=8k and AC=17k. where k is positive.

By Pythagoras theorem, we have,

\[A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}\Rightarrow A{{B}^{2}}=A{{C}^{2}}-B{{C}^{2}}\]

\[\{{{(17k)}^{2}}-{{(5k)}^{2}}\}=(289{{k}^{2}}-64{{k}^{2}})=225{{k}^{2}}\]

\[\Rightarrow \ \ \ AB\sqrt{225{{k}^{2}}}=15k\]

So, \[\cos A=\frac{AB}{AC}=\frac{15k}{17k}=\frac{15}{17},\ \tan A=\frac{BC}{AB}=\frac{8k}{15k}=\frac{8}{15}\]

\[\therefore \] Assertion true; Reason False.