Answer

Verified

394.2k+ views

**Hint:**Here we are given two sides and the angle of a triangle. So we can use the cosine rule to find the equation. Substituting the given data and simplifying we get the required equation.

**Formula used:**The cosine rule states that the square of the length of any side of a triangle equals the sum of the squares of the length of the other sides minus twice the product multiplied by the cosine of their included angle.

In symbols we have,

${a^2} = {b^2} + {c^2} - 2bc\cos A$

Or it can be also written as $\cos A = \dfrac{{{b^2} + {c^2} - {a^2}}}{{2bc}}$

**Complete step-by-step solution:**

Given that in the triangle $ABC$, $a = 4,b = 3,\angle A = 60^\circ $.

We need to find the equation with $c$ as its root.

Here by conventional notation, $a,b,c$ represents the sides opposite to vertices $A,B,C$ respectively.

So we are given two sides and the included angle of the triangle.

Therefore we can apply the cosine rule here.

The cosine rule states that the square of the length of any side of a triangle equals the sum of the squares of the length of the other sides minus twice the product multiplied by the cosine of their included angle.

In symbols we have,

${a^2} = {b^2} + {c^2} - 2bc\cos A$

Or it can be also written as $\cos A = \dfrac{{{b^2} + {c^2} - {a^2}}}{{2bc}}$

So we can substitute the given data.

This gives,

$\Rightarrow$$\cos 60^\circ = \dfrac{{{3^2} + {c^2} - {4^2}}}{{2 \times 3 \times c}}$

Simplifying we get,

$\Rightarrow$$\cos 60^\circ = \dfrac{{9 + {c^2} - 16}}{{6c}}$

$ \Rightarrow \cos 60^\circ = \dfrac{{{c^2} - 7}}{{6c}}$

We know, $\cos 60^\circ = \dfrac{1}{2}$.

Substituting this we get,

$\Rightarrow$$\dfrac{1}{2} = \dfrac{{{c^2} - 7}}{{6c}}$

Cross-multiplying we get,

$\Rightarrow$$2({c^2} - 7) = 6c$

Dividing both sides by $2$ we have,

$\Rightarrow$${c^2} - 7 = 3c$

$ \Rightarrow {c^2} - 3c - 7 = 0$

Thus we get the equation for which $c$ is the root.

**$\therefore $ The answer is option A.**

**Note:**Cosine rule can be used in cases if three sides are given or two sides and the included angle are given. There are three variables in the equation: three sides and one angle. So knowing three of them, we can solve for the fourth.

Recently Updated Pages

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry 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