Answer

Verified

394.8k+ views

**Hint:**We use the sine law of triangle $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$ and given condition from the question to make the equation $\dfrac{\sin A}{\cos A}=\dfrac{\sin B}{\cos B}=\dfrac{\sin C}{\cos C}$. We take first two terms of the equation and use the angle difference e formula of sine $\sin \left( \alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta $ to find relation between $A,B$. We follow the same procedure for the second two terms and find a relation between $B,C$. We find a relation between $A,B,C$ to choose the correct option. \[\]

**Complete step by step answer:**

We know from sine difference of formula that for two angles with measures $\alpha ,\beta $ we have,

\[\sin \left( \alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta \]

We draw the figure of the triangle ABC and denote the measure of the angle subtended at vertices A,B,C are given as $A,B,C$ respectively and length of the sides opposite to vertices A,B,C as $a,b,c$. We have the rough figure as,\[\]

We know from sine law which states that sides of a triangle are proportional to sin of the angle opposite to them which means

\[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\]

Let us assume for some constant $k$ ,

\[\begin{align}

& \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=k \\

& \Rightarrow a=k\sin A,b=k\sin B,c=k\sin C.......\left( 1 \right) \\

\end{align}\]

We are given in the question that the sides of a triangle are proportional to the cosines of the opposite angles. So we have

\[\dfrac{a}{\cos A}=\dfrac{b}{\cos B}=\dfrac{c}{\cos C}\]

We put the values of $a,b,c$ obtain din equation (1) in the above proportion to have,

\[\begin{align}

& \dfrac{k\sin A}{\cos A}=\dfrac{k\sin B}{\cos B}=\dfrac{k\sin C}{\cos C} \\

& \Rightarrow \dfrac{\sin A}{\cos A}=\dfrac{\sin B}{\cos B}=\dfrac{\sin C}{\cos C}.......\left( 2 \right) \\

\end{align}\]

We take first two terms in the above proportion (2) and cross-multiply to have,

\[\begin{align}

& \dfrac{\sin A}{\cos A}=\dfrac{\sin B}{\cos B} \\

& \Rightarrow \sin A\cos B=\cos A\sin B \\

& \Rightarrow \sin A\cos B-\cos A\sin B=0 \\

\end{align}\]

We use sine difference of angle formula for $\alpha =A,\beta =B$ in the above step to have,

\[\begin{align}

& \Rightarrow \sin \left( A-B \right)=0 \\

& \Rightarrow A-B=0 \\

& \Rightarrow A=B......\left( 3 \right) \\

\end{align}\]

We take first second terms in the above proportion (2) and cross-multiply to have,

\[\begin{align}

& \dfrac{\sin B}{\cos B}=\dfrac{\sin C}{\cos C} \\

& \Rightarrow \sin B\cos C=\cos B\sin C \\

& \Rightarrow \sin B\cos C-\cos B\sin C=0 \\

\end{align}\]

We use sine difference of angle formula for $\alpha =B,\beta =C$ in the above step to have,

\[\begin{align}

& \Rightarrow \sin \left( B-C \right)=0 \\

& \Rightarrow B-C=0 \\

& \Rightarrow B=C.......\left( 4 \right) \\

\end{align}\]

We have from equation (3) and (4)

\[A=B=C\]

We know that only equilateral triangles have all three angles equal.

**So, the correct answer is “Option B”.**

**Note:**We note that the general solution of the equation $\sin x=0$ with arbitrary integer $n$ are given by $x=n\pi $ but when we calculated $A-B=0,B-C=0$ we have taken $n=0$ and rejected all other values of $n$ because the difference of two angles cannot be greater than or equal to $\pi $ in a triangle. We must be careful of the confusion between the angle difference formula of sine and cosine which is $\cos \left( \alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta $.

Recently Updated Pages

How do you find slope point slope slope intercept standard class 12 maths CBSE

How do you find B1 We know that B2B+2I3 class 12 maths CBSE

How do you integrate int dfracxsqrt x2 + 9 dx class 12 maths CBSE

How do you integrate int left dfracx2 1x + 1 right class 12 maths CBSE

How do you find the critical points of yx2sin x on class 12 maths CBSE

How do you find the general solution to dfracdydx class 12 maths CBSE

Trending doubts

The provincial president of the constituent assembly class 11 social science CBSE

Gersoppa waterfall is located in AGuyana BUganda C class 9 social science CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The hundru falls is in A Chota Nagpur Plateau B Calcutta class 8 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE