Ex.8.4 Q3 Introduction to Trigonometry Solution - NCERT Maths Class 10


Question

Evaluate

(i) \( \,\,\frac{{{\sin }^{2}}{{63}^{{}^\circ }}+{{\sin }^{2}}{{27}^{{}^\circ }}}{{{\cos }^{2}}{{17}^{{}^\circ }}+{{\cos }^{2}}{{73}^{{}^\circ }}} \)

(ii) \(\,\,\,\text{sin}{{25}^{^\circ }}\text{cos}{{65}^{^\circ }}+\text{cos}{{25}^{^\circ }}\text{ sin}{{65}^{^\circ }} \)

 Video Solution
Introduction To Trigonometry
Ex 8.4 | Question 3

Text Solution

 

Reasoning:

\[\begin{align}{\sin ^{2} A+\cos ^{2} A=1} \\ {\sin \left(90^{\circ}-\theta\right)=\cos \theta} \\ {\cos \left(90^{\circ}-\theta\right)=\sin \theta}\end{align}\]

Steps:

(i)\(\begin{align}\frac{{{\sin }^{2}}{{63}^{{}^\circ }}+{{\sin }^{2}}{{27}^{{}^\circ }}}{{{\cos }^{2}}{{17}^{{}^\circ }}+{{\cos }^{2}}{{73}^{{}^\circ }}}\end{align}\)

\[\begin{align} & =\frac{{{\left[ \sin \left( {{90}^{{}^\circ }}-27 \right) \right]}^{2}}+{{\sin }^{2}}27}{{{\left[ \cos \left( {{90}^{{}^\circ }}-{{73}^{{}^\circ }} \right) \right]}^{2}}+{{\cos }^{2}}{{73}^{{}^\circ }}} \\ & =\frac{{{[\cos 27]}^{2}}+{{\sin }^{2}}{{27}^{{}^\circ }}}{{{\left[ \sin {{73}^{{}^\circ }} \right]}^{2}}+{{\cos }^{2}}{{73}^{{}^\circ }}}
\\ & \begin{bmatrix}  \sin \left( {{90}^{{}^\circ }}-\text{ }\!\!\theta\!\!\text{ } \right)=\cos \text{ }\!\!\theta\!\! \\ \And \\ \cos \left( {{90}^{{}^\circ }}-\theta \right)=\sin \theta  \end{bmatrix} \\ & =\frac{1}{1} \; \begin{bmatrix} \text{By identity} \\ \sin ^{2} A+\cos ^{2} A=1 \end{bmatrix} \\& =1\end{align}\]

(ii) \(\sin {{25}^{{}^\circ }}\cos {{65}^{{}^\circ }}+\cos {{25}^{{}^\circ }}\sin {{65}^{{}^\circ }}\)

\[\begin{align} &= \left( \sin {{25}^{{}^\circ }} \right)\left\{ \cos \left( {{90}^{{}^\circ }}-{{25}^{{}^\circ }} \right) \right\} +\cos {{25}^{{}^\circ }} \left\{ \sin \left( {{90}^{{}^\circ }}-{{25}^{{}^\circ }} \right) \right. \\ & \begin{bmatrix} \sin \left( {{90}^{{}^\circ }}-\text{ }\!\!\theta\!\!\text{ } \right)=\cos \text{ }\!\!\theta\!\! \\ \And  \cos \left( {{90}^{{}^\circ }}-\theta \right)=\sin \theta \end{bmatrix} \\ & = \begin{pmatrix} \left( \sin {{25}^{{}^\circ }} \right)\left( \sin {{25}^{{}^\circ }} \right)  +\cos {{25}^{{}^\circ }}\left( \cos {{25}^{{}^\circ }} \right) \end{pmatrix}\\ & ={{\sin }^{2}}{{25}^{{}^\circ }}+{{\cos }^{2}}{{25}^{{}^\circ }} \\ & =1 \\ \\& \begin{bmatrix} \text{By identity} \\ \sin ^{2} A+\cos ^{2} A=1 \end{bmatrix} \end{align}\]

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