Question

if tanA= 3/4 find the value of cotA , secA, cosecA​

Answer 1

Answer:

Since,tanA will be +ve only in 1st $3rd quadrant, Hence possible value of CosecA=+5/3 or - 5/3 and secA=+5/4 or -5/4 , because in 3rd quadrant cosec and sec are negative..

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Answer 2

$\maltese \textbf{ \: \: if tanA= }{\dfrac{3}{4} \: \: \textbf{find the value of cotA , secA, cosecA}}$

$\qquad \qquad \qquad \huge{{ \frak{Solution}}}$

$\textbf {given \: that}\: \: \tt: \: tan A \: \dfrac{3}{4} \\ \\ \\ \sf: \longmapsto \frac{ \textsf{Side opposite to angle A}}{ \textsf{Side odjucent to angleA}} = \frac{3}{4} \\ \\ \sf: \longmapsto \frac{BC }{AB } = \frac{3}{4} \\ \\ \bf \: let \: \: \: \frac{BC}{ AB } = \frac{3x}{4x } \\ \\ \\ \maltese \underline{ \textbf{We need to Find AC so we use Pythagoras theorem}} \\ \\ \textbf{In right triangle ABC} \\ \\ \green{ \bf using \:phythagoras \: theorem } \\ \\ \sf: \dashrightarrow \underline{ \pink{ \boxed{ \pmb{(Hypotenuse)² = (Height)²+(Base)²}}}} \\ \\ \sf: \dashrightarrow(AC)² =(BC)²+(AB)² \\ \\ \sf: \dashrightarrow(AC)² = {(3x)}^{2} + {(4x)}^{2} \\ \\ \sf: \dashrightarrow(AC)² = 9x + 16x \\ \\ \sf: \dashrightarrow(AC)² = 25x ^{2} \\ \\ \sf \: now \: taking \: the \: square \: root \: to \: both \: side \: \\ \\ \sf: \dashrightarrow(AC) = \sqrt{25 {x}^{2} } \\ \\ \sf: \dashrightarrow \underline{ \boxed{ \sf(AC) = 5x}} \bigstar \\ \\ \\ \large \bf \: now \: \\ \\ \\ \underline{ \boxed{ \bf \: sin A = \dfrac{ side \: opposite \: to \: angel \: A }{Hypotenuse }}} \\ \\ \sf: \dashrightarrow \: sin A = \dfrac{BC}{AC} \\ \\ \sf: \dashrightarrow \: sin A = \frac{3x}{5x} \: \\ \\ \sf: \dashrightarrow \: sin A = \frac{3}{5} \\ \\ \\ \large \bf \: similarly \\ \\ \\ \underline{ \boxed{ \bf \: cosA = \dfrac{side \: adjacent \: to \: A }{Hypotebuse}}} \\ \\ \sf: \dashrightarrow \: cosA = \dfrac {AB}{AC} \\ \\ \sf: \dashrightarrow \: cosA = \frac{3x}{5x} \\ \\ \sf: \dashrightarrow \: cosA = \frac{3}{5}$

$\\ \large \bf \: given =tan A = \frac{3}{4}$

$\qquad \qquad \maltese \underline{ \boxed{ \bf \: cosec A = \dfrac{1}{sinA} }} \\ \\ \sf \: cosec A = \frac{1}{4/5} = \frac{4}{5}$

$\qquad \qquad \maltese \underline{ \boxed{ \bf \: sec A = \dfrac{1}{cosA} }} \\ \\ \sf \: sec A = \frac{1}{3/5} = \frac{5}{3}$

$\qquad \qquad \maltese \underline{ \boxed{ \bf \: cotA = \dfrac{1}{tanA} }} \\ \\ \sf \: cot A = \frac{1}{3/4} = \frac{4}{3}$

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$\\ \\ \underline{ \boxed{ \textbf{The value of} \sf \: \: cotA = \frac{4}{3} , secA = \frac{5}{3} , cosecA = \frac{5}{4} }}$