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Sine and Cosine Formulae and Their Applications — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSine and Cosine Formulae and Their Applications1 Mark
Question
If in a $\triangle\text{ABC},\frac{\cos\text{A}}{\text{a}}=\frac{\cos\text{B}}{\text{b}}=\frac{\cos\text{C}}{\text{c}},$ then find the measures of angles A, B, C.
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Answer

$\frac{\cos\text{A}}{\text{a}}=\frac{\cos\text{B}}{\text{b}}=\frac{\cos\text{C}}{\text{c}}$ $\Rightarrow\frac{\text{b}^2+\text{c}^2-\text{a}^2}{2\text{abc}}=\frac{\text{a}^2+\text{c}^2-\text{b}^2}{2\text{abc}}=\frac{\text{a}^2+\text{b}^2-\text{c}^2}{2\text{abc}}$ $\Rightarrow\text{b}^2+\text{c}^2-\text{a}^2=\text{a}^2+\text{c}^2-\text{b}^2=\text{a}^2+\text{b}^2-\text{c}^2$ $\therefore\text{b}^2+\text{c}^2-\text{a}^2=\text{a}^2+\text{c}^2-\text{b}^2$ $2\text{b}^2=2\text{a}^2\Rightarrow\text{b = a}$ $\therefore\text{a}^2+\text{c}^2-\text{b}^2=\text{a}^2+\text{b}^2-\text{c}^2$ $2\text{c}^2=2\text{b}^2\Rightarrow\text{c = b}$ $\text{b}^2+\text{c}^2-\text{a}^2=\text{a}^2+\text{b}^2-\text{c}^2$ $\Rightarrow2\text{c}^2=2\text{a}^2\Rightarrow\text{c = a}$ Hence $\text{a = b = c}$ or, $\text{A = B = C}=60^{\circ}$

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