
The number of triangles ABC that can be formed with a=3,b=8 and $\sin A=\dfrac{5}{13}$ is [Roorkee Qualifying 1998]
A. 0
B. 1
C. 2
D. 3
Answer
164.7k+ views
Hint:
In this question, we are provided with the two sides and sine measurement of one angle. In order to find the number of triangles formed we will apply the law of the sines formula, relating the given lengths of the sides of the triangle to the sines of their consecutive angles.
Formula Used:
Laws of Sines for triangle $ABC$ whose length is $a, b$, and $c$ respectively is given by;
$\dfrac{a}{\sin A}= \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$
$\dfrac{\sin A}{a} = \dfrac{\sin B}{b} =\dfrac{\sin C}{c}$
Complete step-by-step solution:
It is given that in the triangle $ABC$, $a=3,b=8$ and $\sin A=\dfrac{5}{13}$.
Applying the law of sines formula we get;
$\dfrac{\sin B}{b}=\dfrac{\sin A}{a}$
$\Rightarrow\dfrac{\sin B}{8}=\dfrac{5/13}{3}$
$\Rightarrow sin B=\dfrac{8}{3}\times\dfrac{5}{13}$
\[\Rightarrow \sin B=\dfrac{40}{39}>1\]
Since the value is greater than $1$ it is not possible. So, no triangle is possible with the $a=3,b=8$ and $\sin A=\dfrac{5}{13}$.
So, option A is correct.
Note:
The alternative way to find the numbers of triangles formed we can use the law of cosines. Both the law of sines and cosines are applicable in finding the unknown values of the angle or an unknown side of a given triangle. Laws of cosine for triangle $ABC$ whose length is $a, b$, and $c$ respectively is given by;
$a^2 = b^2 + c^2 − 2bc.\cos A$
$b^2 = a^2 +c^2 − 2ac.\cos B$
$c^2 = a^2 + b^2 − 2ab.\cos C$.
In this question, we are provided with the two sides and sine measurement of one angle. In order to find the number of triangles formed we will apply the law of the sines formula, relating the given lengths of the sides of the triangle to the sines of their consecutive angles.
Formula Used:
Laws of Sines for triangle $ABC$ whose length is $a, b$, and $c$ respectively is given by;
$\dfrac{a}{\sin A}= \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$
$\dfrac{\sin A}{a} = \dfrac{\sin B}{b} =\dfrac{\sin C}{c}$
Complete step-by-step solution:
It is given that in the triangle $ABC$, $a=3,b=8$ and $\sin A=\dfrac{5}{13}$.
Applying the law of sines formula we get;
$\dfrac{\sin B}{b}=\dfrac{\sin A}{a}$
$\Rightarrow\dfrac{\sin B}{8}=\dfrac{5/13}{3}$
$\Rightarrow sin B=\dfrac{8}{3}\times\dfrac{5}{13}$
\[\Rightarrow \sin B=\dfrac{40}{39}>1\]
Since the value is greater than $1$ it is not possible. So, no triangle is possible with the $a=3,b=8$ and $\sin A=\dfrac{5}{13}$.
So, option A is correct.
Note:
The alternative way to find the numbers of triangles formed we can use the law of cosines. Both the law of sines and cosines are applicable in finding the unknown values of the angle or an unknown side of a given triangle. Laws of cosine for triangle $ABC$ whose length is $a, b$, and $c$ respectively is given by;
$a^2 = b^2 + c^2 − 2bc.\cos A$
$b^2 = a^2 +c^2 − 2ac.\cos B$
$c^2 = a^2 + b^2 − 2ab.\cos C$.
Recently Updated Pages
Environmental Chemistry Chapter for JEE Main Chemistry

Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

Get P Block Elements for JEE Main 2025 with clear Explanations

Sets, Relations and Functions Chapter For JEE Main Maths

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

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

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

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

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

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

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics
