Question

if sin theta + cosec theta = 10/3. find the value of sin square theta + cosec square theta ​

Answer 1

Question:

$\tt If ~sin \theta + cosec \theta = \frac{10}{3}. Find ~the ~value ~of ~sin^2 \theta + cosec^2 \theta$

Solution:

$\tt sin \theta + cosec \theta = \frac{10}{3} \\\\ \tt Squaring~on~both~sides~: \\\\ \bigg(sin \theta + cosec \theta \bigg)^2= \bigg(\frac{10}{3}\bigg)^2 \\\\ \tt Using~ \boxed{(a+b)^2=(a^2+b^2+2ab)} ,~we~have \\\\ \tt \to sin^2 \theta + cosec^2 \theta +2 sin \theta cosec \theta = \frac{100}{9} \\\\ \to \tt sin^2 \theta + cosec^2 \theta +2 sin \theta \frac{1}{sin\theta} = \frac{100}{9} \\\\\to \tt sin^2 \theta + cosec^2 \theta +2 \cancel{sin \theta} \frac{1}{\cancel{sin\theta}} = \frac{100}{9} \\\\ \to\tt sin^2 \theta + cosec^2 \theta = \frac{100}{9}-2 \\\\\to \tt sin^2 \theta + cosec^2 \theta = \frac{100-18}{9} \\\\ \to\tt sin^2 \theta + cosec^2 \theta = \frac{82}{9}$

Explore more:

$\begin{gathered}\sf Reciprocal \: Relation\\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \boxed{\boxed{\begin{array}{ c|}\rm Sin \theta = & \frac{1}{Cosec\theta}\\ \\ \rm Cos \theta = & \frac{1}{Sec \theta} \\ \\ \rm Tan\theta = & \frac{1}{Cot\theta}\\ \\ \rm cosec\theta = & \frac{1}{Sin \theta} \\ \\ \rm Sec\theta = & \frac{1}{Cos\theta} \\ \\ \rm Cot \theta = & \frac{1}{Tan\theta}\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}$

$\begin{gathered}\sf Trigonometric\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}$