
The smallest angle of the ABC when $a=7,b=4\sqrt{3} and c=\sqrt{13}$, is [MP PET 2003]
A. ${{30}^{o}}$
B. ${{15}^{o}}$
C. ${{45}^{o}}$
D. None of these
Answer
162k+ views
Hint: In this question, we have given the angles of a triangle with its side measures. To find the smallest angle use the laws of the cosine of angle relation. Use rationalization to simplify the expression. Lastly, compare the obtained result with the option provided.
Formula used:Laws of cosine for triangle $ABC$ whose length is $a, b$, and $c$ respectively is given by;
$b^2 = a^2 +c^2 − 2ac.\cos B$.
Complete step by step solution:Given sides of $\Delta ABC$ are $a=7,b=4\sqrt{3}$ and $c=\sqrt{13}$.
We know that the smallest angle in a triangle has the shortest opposing side and here the shortest side is $c=\sqrt{13}$.
Therefore,
$\cos C=\dfrac{{{b}^{2}}+{{a}^{2}}-{{c}^{2}}}{2ab}$
Substituting the values we get;
$=\dfrac{(7)^2+(4\sqrt{3})^2-(\sqrt{13})^2}{2\times 7\times 4\sqrt{3}}
\\=\dfrac{49+48-13}{2\times 7\times 4\sqrt{3}}\\
=\dfrac{84}{56\sqrt{3}}\\
=\dfrac{3}{2\sqrt{3}}$
Now rationalize;
$=\dfrac{3}{2\sqrt{3}}\times \dfrac{2\sqrt{3}}{2\sqrt{3}}\\
=\dfrac{6\sqrt{3}}{12}\\
=\dfrac{\sqrt{3}}{2}$
Thus, $\cos C=\dfrac{\sqrt{3}}{2}$
$\Rightarrow C=\cos^{-1} \dfrac{\sqrt{3}}{2}$
$\Rightarrow C={{30}^{o}}$
Thus, the smallest angle is
$\angle C={{30}^{o}}$
Thus, Option (A) is correct.
Note: Keep in mind that here we are finding the smallest angle in a triangle and we know that the smallest angle in a triangle has the shortest opposing side. Here it is C. Using the law of cosine for angle C, students usually make mistakes in taking the wrong angles. Different angles use cosine rules in a different way.
Formula used:Laws of cosine for triangle $ABC$ whose length is $a, b$, and $c$ respectively is given by;
$b^2 = a^2 +c^2 − 2ac.\cos B$.
Complete step by step solution:Given sides of $\Delta ABC$ are $a=7,b=4\sqrt{3}$ and $c=\sqrt{13}$.
We know that the smallest angle in a triangle has the shortest opposing side and here the shortest side is $c=\sqrt{13}$.
Therefore,
$\cos C=\dfrac{{{b}^{2}}+{{a}^{2}}-{{c}^{2}}}{2ab}$
Substituting the values we get;
$=\dfrac{(7)^2+(4\sqrt{3})^2-(\sqrt{13})^2}{2\times 7\times 4\sqrt{3}}
\\=\dfrac{49+48-13}{2\times 7\times 4\sqrt{3}}\\
=\dfrac{84}{56\sqrt{3}}\\
=\dfrac{3}{2\sqrt{3}}$
Now rationalize;
$=\dfrac{3}{2\sqrt{3}}\times \dfrac{2\sqrt{3}}{2\sqrt{3}}\\
=\dfrac{6\sqrt{3}}{12}\\
=\dfrac{\sqrt{3}}{2}$
Thus, $\cos C=\dfrac{\sqrt{3}}{2}$
$\Rightarrow C=\cos^{-1} \dfrac{\sqrt{3}}{2}$
$\Rightarrow C={{30}^{o}}$
Thus, the smallest angle is
$\angle C={{30}^{o}}$
Thus, Option (A) is correct.
Note: Keep in mind that here we are finding the smallest angle in a triangle and we know that the smallest angle in a triangle has the shortest opposing side. Here it is C. Using the law of cosine for angle C, students usually make mistakes in taking the wrong angles. Different angles use cosine rules in a different way.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2026 Syllabus PDF - Download Paper 1 and 2 Syllabus by NTA

JEE Main Eligibility Criteria 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
