Sine Rule MCQ Quiz - Objective Question with Answer for Sine Rule - Download Free PDF
Last updated on Mar 17, 2025
Latest Sine Rule MCQ Objective Questions
Sine Rule Question 1:
The sides of a triangle are in \(A.P\), and the greatest angle is double of smallest angle. Then the sum of ratio of its sides is
Answer (Detailed Solution Below) 15
Sine Rule Question 1 Detailed Solution
Calculation
Let the sides be \(a-d, a, a+d\).
It is understood that \(a > d > 0\) and from the figure, \(\angle C\) is greatest and \(\angle A\) is smallest.
By given condition, \(C = 2A\).
And hence \(B = \pi - (A + C) = \pi - 3A\).
Hence by sine rule we have,
\(\dfrac{a+d}{\sin C} = \dfrac{a-d}{\sin A} = \dfrac{a}{\sin B}\)
or \(\dfrac{a+d}{\sin 2A} = \dfrac{a-d}{\sin A} = \dfrac{a}{\sin(\pi - 3A)}\).
\(\therefore 2\cos A = \dfrac{a+d}{a-d}\) and \(\dfrac{a}{a-d} = \dfrac{\sin 3A}{\sin A}\).
\(\therefore \dfrac{a}{a-d} = 3 - 4\sin^2 A = 3 - 4 + (2\cos A)^2\).
\(\therefore \dfrac{a}{a-d} = 1 + \left(\dfrac{a+d}{a-d}\right)^2 = \dfrac{4ad}{(a-d)^2}\).
\(a \neq 0\) \(\therefore a-d = 4d\) or \(a = 5d\).
\(\therefore\) sides are \(a-d, a, a+d\)
or \(4d, 5d, 6d\).
Hence the required ratio is \(4:5:6\).
Then the sum of ratio of its sides is = 15
Sine Rule Question 2:
In \(\Delta ABC\); with usual notations, if \(\cos A=\dfrac{\sin B}{\sin C}\), then the triangle is _______.
Answer (Detailed Solution Below)
Sine Rule Question 2 Detailed Solution
Thus, \(\cos A = \dfrac{AC}{AB}\).
Using above relation we can claim that \(\triangle ABC\) is a right angled triangle with AB as hypotenuse, right angled at C.
Sine Rule Question 3:
In \(\triangle ABC\) if \(\sin^{2} A+ \sin^{2}B = \sin^{2}C\) and \(l(AB) = 10\), then the maximum value of the area of \(\triangle ABC\) is
Answer (Detailed Solution Below)
Sine Rule Question 3 Detailed Solution
Calculation
\(\sin^{2} A+ \sin^{2}B = \sin^{2}C\)
\(\Rightarrow a^{2} + b^{2} = c^{2}\) (Sine Rule)
\(A(\triangle ABC) = \dfrac {1}{2} ab ..... (1)\)
From sine rule \(\dfrac {a}{\sin A} = \dfrac {b}{\sin B} = \dfrac {c}{\sin C}\)
\(\Rightarrow \dfrac {a}{\sin A} = \dfrac {b}{\sin B} = \dfrac {10}{1}\)
\(\Rightarrow a = 10\sin A, b = 10\sin B\)
Using equation \((1)\)
\(A(\triangle ABC) = \dfrac {1}{2} (10\sin A)(10\sin B)\)
\(= 50\sin A \sin B\)
But maximum value of \(\sin A \sin B = \dfrac {1}{2}\)
\(\therefore\) Maximum value of \(A (\triangle ABC) = 50\times \dfrac {1}{2} = 25\)
OR
\(\angle C = 90^{\circ}\Rightarrow ABC\) is right angled triangle
\(\therefore\) Area of \(\triangle\) is maximum when it is \(45^{\circ} - 45^{\circ} - 90^{\circ}\triangle\).
\(\therefore A(\triangle ABC) = \dfrac {1}{2}\times 5\sqrt {2}\times 5\sqrt {2} = 25\)
Hence option 3 is correct
Sine Rule Question 4:
If one side of a triangle is double the other and the angles opposite to these sides differ by 60°, then the triangle is
Answer (Detailed Solution Below)
Sine Rule Question 4 Detailed Solution
Answer : 2
Solution :
In ΔABC, by sine rule,
\(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\)
According to the given condition,
In ΔABC, a = 2 b and
A - B = 60° A = 60° + B
⇒ \(\frac{\sin \left(60^{\circ}+B\right)}{2 b}=\frac{\sin B}{b}\)
⇒ \(\frac{\sin B}{\sin \left(B+60^{\circ}\right)}=\frac{1}{2}\)
⇒ 2 sin B = sin B cos 60° + cos B sin 60°
⇒ \(2 \sin \mathrm{~B}=\sin \mathrm{B}\left(\frac{1}{2}\right)+\cos \mathrm{B}\left(\frac{\sqrt{3}}{2}\right)\)
⇒ \(\frac{3}{2} \sin \mathrm{~B}=\frac{\sqrt{3}}{2} \cos \mathrm{~B}\)
⇒ \(\tan B=\frac{1}{\sqrt{3}} \Rightarrow B=30^{\circ}\)
∴ A = 30° + 60° = 90°
∴ ΔABC is right angled.
Sine Rule Question 5:
In a triangle ABC, with usual notations, if c = 4, then value of \((a-b)^{2} \cos ^{2} \frac{C}{2}+(a+b)^{2} \sin ^{2} \frac{C}{2}\) is
Answer (Detailed Solution Below)
Sine Rule Question 5 Detailed Solution
Concept Used:
1. Cosine Rule: \(c^2 = a^2 + b^2 - 2ab \cos C\)
2. Half-angle formulas:
\(\cos^2 \frac{C}{2} = \frac{1 + \cos C}{2}\)
\(\sin^2 \frac{C}{2} = \frac{1 - \cos C}{2}\)
Calculation:
Given:
In triangle ABC, c = 4
Expression: \((a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2}\)
⇒ \((a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2}\)
⇒ \((a - b)^2 \left(\frac{1 + \cos C}{2}\right) + (a + b)^2 \left(\frac{1 - \cos C}{2}\right)\)
⇒ \(\frac{1}{2} \left[ (a^2 - 2ab + b^2)(1 + \cos C) + (a^2 + 2ab + b^2)(1 - \cos C) \right]\)
⇒ \(\frac{1}{2} \left[ a^2 + a^2 \cos C - 2ab - 2ab \cos C + b^2 + b^2 \cos C + a^2 - a^2 \cos C + 2ab - 2ab \cos C + b^2 - b^2 \cos C \right]\)
⇒ \(\frac{1}{2} \left[ 2a^2 + 2b^2 - 4ab \cos C \right]\)
⇒ \(a^2 + b^2 - 2ab \cos C\)
⇒ \(c^2\) (using the cosine rule)
⇒ \(4^2\) (since c = 4)
⇒ 16
∴ The value of the expression is 16.
Hence option 2 is correct
Top Sine Rule MCQ Objective Questions
If sin (C + D) = √3/2 and sec (C - D) = 2/√3 then what is the value of C and D?
Answer (Detailed Solution Below)
Sine Rule Question 6 Detailed Solution
Download Solution PDFGiven:
sin (C + D) = √3/2
sec (C - D) = 2/√3
Calculations:
If sin (C + D) = √3/2 and sec (C - D) = 2/√3
Then,
⇒ C + D = 60°.............(1)
⇒ C - D = 30°..............(2)
Solving 1 & 2 .
C = 45°
D = 15°
∴ Option 1 is the correct answer.
If the angle of triangle A, B and C are in AP and b : a = √3 : 1, then what is the value of A?
Answer (Detailed Solution Below)
Sine Rule Question 7 Detailed Solution
Download Solution PDFConcepts:
Sine law ⇔ \(\frac{{\rm{a}}}{{\sin {\rm{A}}}} = {\rm{\;}}\frac{{\rm{b}}}{{\sin {\rm{B}}}} = {\rm{\;}}\frac{{\rm{c}}}{{\sin {\rm{C}}}}\)
Where a, b and c are sides and A, B and C are angles.
Here; Side a faces angle A, side b faces angle B and side c faces angle C
Calculation:
Here, A, B and C are in AP
So, 2B = A + C
And sum of angles of triangle, A + B + C = 180°
⇒ 2B + B = 180
⇒ 3B = 180
⇒ B = 60°
Now, by sine rule,
\(\frac{{\rm{a}}}{{\sin {\rm{A}}}} = {\rm{\;}}\frac{{\rm{b}}}{{\sin {\rm{B}}}} \)
\(\rm ⇒ \frac{b}{a}=\frac{\sin B}{\sin A}\)
\(\rm ⇒ \frac{√3 }{1}=\frac{\sin 60^o}{\sin A}\)
⇒ sin A = (√3/2) × (1/√3)
⇒ sin A = 1/2
⇒ A = 30°
Hence, option (1) is correct.
In a triangle ABC if a = 2, b = 4 and sin A = 1/4, then what is angle B equal to?
Answer (Detailed Solution Below)
Sine Rule Question 8 Detailed Solution
Download Solution PDFConcept:
Sine Rule:
\(\rm \frac{sin\: A}{a} = \frac{sin\: B}{b} = \frac{sin\: C}{c} \)
Where a, b and c are sides and A, B and C are angles.
Here; Side a faces angle A, side b faces angle B and side c faces angle C
Calculation:
Given: a = 2, b = 4 and sin A = 1/4
\(\rm \frac{sin\: A}{a} = \frac{sin\: B}{b}\)
⇒ \(\rm \frac{1/4}{2} = \frac{sin\: B}{4}\)
⇒ sin B = \(\rm 1\over2\)
⇒ B = 30∘ = \(\rm \pi\over6\)
Additional Information
Cosine Rule:
cos A = \(\rm \frac{b^{2} + c^{2} - a^{2}}{2bc}\)
cos B = \(\rm \frac{c^{2} + a^{2} - b^{2}}{2ac}\)
cos C = \(\rm \frac{a^{2} + b^{2} - c^{2}}{2ab}\)
In ΔABC if sin2 A + sin2 B = sin2 C and \(l (AB) =10\) then the maximum value of the area of ΔABC is
Answer (Detailed Solution Below)
Sine Rule Question 9 Detailed Solution
Download Solution PDFConcept:
In ΔABC, sine rule
\(\rm \dfrac {a}{sin\; A} = \dfrac {b}{sin\; B} = \dfrac {c}{sin\;C} \)
\(\rm A (ΔABC) = \dfrac{1}{2}\times base \;\times height \)
Calculations:
Given, In ΔABC if sin2 A + sin2 B = sin2 C and
length of Side (AB) = 10 ⇒ c = 10
⇒ a2 + b2 = c2
Pythagoreans theorem is verified.
\(\rm \angle C = 90^\circ\)
Hence, ΔABC is right angled triangle.
\(\rm A (ΔABC) = \dfrac{1}{2}\times base \;\times height \)
\(\rm A (ΔABC) = \dfrac{1}{2}\times a \times b\)....(1)
By sine rule, we have
\(\rm \dfrac {a}{sin\; A} = \dfrac {b}{sin\; B} = \dfrac {c}{sin\;C} \)
⇒\(\rm \dfrac {a}{sin\; A} = \dfrac {b}{sin\; B} = \dfrac {10}{sin\; 90^\circ} \)
⇒\(\rm \dfrac {a}{sin\; A} = \dfrac {b}{sin\; B} = \dfrac {10}{1} \)
⇒\(\rm a = 10\;sin\;A\) and \(\rm b = 10\;sin\;B\)
Equation (1) becomes,
\(\rm A (ΔABC) = \dfrac{1}{2}\times 10 \;sin\; A \times 10\;sin\; B \)
⇒\(\rm A (ΔABC) = 50 \times sin\; A \times sin\; B \) ....(2)
Since, \(\rm \angle C = 90^\circ\)and sum of all angles of triangle is 180º
\(\rm \angle (A+B) = 90^\circ\)
⇒Max (\(\rm sin\; A \times sin\; B\)) = \(\rm \dfrac 1 2\)
Equation (2) becomes,
\(\rm A (ΔABC) = 50 \times \dfrac 1 2 \)
\(\rm A (ΔABC) = 25 \)
Consider the following statements:
1. If ABC is a right-angled triangle, right-angled at A, and if sin \(\rm B = \frac 1 3,\) then cosec C = 3.
2. If b cos B = c cos C and if the triangle ABC is not right-angled, then ABC must be isosceles.
Which of the above statements is/are correct?
Answer (Detailed Solution Below)
Sine Rule Question 10 Detailed Solution
Download Solution PDFConcept:
Pythagoras theorem: In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.
Sine Rule in triangle ABC, having sides a, b, c:
\(\rm \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{K}\)
sin A - cos B = \(\rm 2\cos(\frac{A +B}{2})\sin(\frac{A -B}{2})\)
Calculation:
1. We have,
sin \(\rm B = \frac 1 3=\frac{P}{H}\)
Hwence, Right angle triangle can be drawn as,
Here, P = 1
H = 3
B = x
Using Pythagoras theorem,
32 = 12 + x2
x2 = 8
\(\therefore x=2\sqrt{2}\)
Now , Cosec C = \(\frac{3}{2\sqrt{2}}\)
Hence Ststement (1) is not correct.
2. Given,
b cos B = c cos C
Using the sine rule,
\(\rm \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{K}\)
⇒ b = K sin B & c = K sin C
⇒ 2 sin B cos B = 2 sin C cos C
Assume,
K = 2
⇒ 2 sin B cos B = 2 sin C cos C
⇒ sin 2B = sin 2C [2 sin x cos x = sin 2x]
⇒ sin 2B - sin 2C = 0
Using Formula:
sin C - sin D = 2 cos \((\frac{C+D}{2})\) × sin \((\frac{C-D}{2})\)
⇒ 2 cos (B + C) sin (B - C) = 0
In this case,
Either,
cos (B + C) = 0
Hence, (B + C) = 90° .... (1)
Or,
Or sin (B -C) = 0
B - C = 0
⇒ B = C .... (2)
From equation (1) & (2),
B = C = 45°
So ABC must be issosceles
Hence, option (2) is correct.
In a triangle ABC, side c = 2, angle A = 45°, side \({\rm{a}} = 2\sqrt 2 \), then what is angle C equal to?
Answer (Detailed Solution Below)
Sine Rule Question 11 Detailed Solution
Download Solution PDFConcept:
In triangle ABC, According to Sine rule
\(\frac{a}{\text{sinA}}=\frac{b}{\text{sinB}}=\frac{c}{\text{sinC}}\) Where, a, b, c are sides and A, B, C are angles of triangle.
Calculation:
In a triangle ABC, c = 2, A = 45°, \({\rm{a}} = 2\sqrt 2 \),
Using sine rule in given triangle,
\(\frac{a}{\text{sinA}}=\frac{c}{\text{sinC}}\\ \Rightarrow \frac{2\sqrt2}{\text{sin45}}=\frac{2}{\text{sinC}}\\\Rightarrow \text{sinC} = \frac{\text{sin45}}{\sqrt2}=\frac{1}{\sqrt2\sqrt2}\\ \Rightarrow \text{sinC} =\frac{1}{2}\\ \Rightarrow C =30\)
Hence, option (1) is correct.
If any ΔABC , ∠C = 75°, ∠B = 45° and a = √3, then find the value of b.
Answer (Detailed Solution Below)
Sine Rule Question 12 Detailed Solution
Download Solution PDFConcept:
Concept:
In triangle ABC, According to the Sine rule
\(\rm \frac{a}{\text{sin A}}=\frac{b}{\text{sin B}}=\frac{c}{\text{sin C}}\)
Where, a, b, c are sides and A, B, C are angles of triangle.
Sum of the angle in triangle is 180°
Calculation:
Given: ∠C = 75°, ∠B = 45° and a = √3
∠A + ∠B + ∠C = 180°
⇒ ∠A = 180° - 75° - 45°
⇒ ∠A = 60°
Sine rule
\(\frac{a}{\text{sinA}}=\frac{b}{\text{sinB}}\)
⇒ \(\rm \frac {√3}{sin\ 60^0}\ = \ \frac {b}{sin\ 45^0}\)
⇒ \(\rm \frac {√3}{\frac {√3}{2}}\ = \ \frac {b}{\frac {1}{√2}}\)
⇒ \(\rm b\ = \frac {2}{√2}\)
⇒ b = √2
In a triangle ABC, a = (1 + √3) cm, b = 2 cm and angle C = 60°, then the other two angles are
Answer (Detailed Solution Below)
Sine Rule Question 13 Detailed Solution
Download Solution PDFConcept:
Consider a triangle ABC,
Cosine rule:
\(\cos {\rm{C}} = \frac{{{{\rm{a}}^2} + {{\rm{b}}^2} - {{\rm{c}}^2}}}{{2{\rm{ab}}}}\)
Sine rule:
\(\frac{{\rm{a}}}{{\sin {\rm{A}}}} = \frac{{\rm{b}}}{{\sin {\rm{B}}}} = \frac{{\rm{c}}}{{\sin {\rm{C}}}}\)
Calculation:
Given: a = (1 + √ 3) cm, b = 2 cm and ∠C = 60°
We know that, \(\cos {\rm{C}} = \frac{{{{\rm{a}}^2} + {{\rm{b}}^2} - {{\rm{c}}^2}}}{{2{\rm{ab}}}}\)
\( \Rightarrow \cos 60^\circ = \frac{{{{\left( {1 + \sqrt 3 } \right)}^2} + {2^2} - {{\rm{c}}^2}}}{{2\left( {1 + \sqrt 3 } \right)\left( 2 \right)}}\)
⇒ 2(1 + √3) = 1 + 3 + 2√3 + 4 – c2
⇒ c2 = 6
⇒ c = √6
Using sin rule we get,
\(\frac{{\rm{a}}}{{\sin {\rm{A}}}} = \frac{{\rm{b}}}{{\sin {\rm{B}}}} = \frac{{\rm{c}}}{{\sin {\rm{C}}}}\)
\( \Rightarrow \frac{{1 + \sqrt 3 }}{{\sin {\rm{A}}}} = \frac{2}{{\sin {\rm{B}}}} = \frac{{\sqrt 6 }}{{\sin 60^\circ }}\)
⇒ sin B = (2 / √6) × sin 60°
\( \Rightarrow \sin {\rm{B}} = \frac{1}{{\sqrt 2 }}\)
⇒ B = 45°
Now, sum of all angles of triangles = 180°
⇒ A + B + C = 180°
⇒ A = 180° – 45° – 60°
⇒ A = 75°
Hence, other angles are 45° and 75°.In a triangle ABC if a = 3, b = 4 and sin A = 3/4, then what is angle B equal to?
Answer (Detailed Solution Below)
Sine Rule Question 14 Detailed Solution
Download Solution PDFConcept:
Sine Rule:
\(\rm \frac{sin\: A}{a} = \frac{sin\: B}{b} = \frac{sin\: C}{c} \)
Where a, b and c are sides and A, B and C are angles.
Here; Side a faces angle A, side b faces angle B and side c faces angle C
Calculation:
Given:
a = 3, b = 4 and sin A = 3/4
\(\rm \frac{sin\: A}{a} = \frac{sin\: B}{b}\)
\(\rm \frac{3/4}{3} = \frac{sin\: B}{4}\)
sin B = 1
B = 90∘ = π/2
Additional Information
Cosine Rule:
cos A = \(\rm \frac{b^{2} + c^{2} - a^{2}}{2bc}\)
cos B = \(\rm \frac{c^{2} + a^{2} - b^{2}}{2ac}\)
cos C = \(\rm \frac{a^{2} + b^{2} - c^{2}}{2ab}\)
Consider the following statements:
1. There exists no triangle ABC for which sin A + sin B = sin C.
2. If the angles of a triangle are in the ratio 1 : 2 : 3, then its sides will be in the ratio 1 : √3 : 2.
Which of the above statements is/are correct?Answer (Detailed Solution Below)
Sine Rule Question 15 Detailed Solution
Download Solution PDFConcept:
\(\rm \frac{a}{\ sin\;A}=\frac{b}{sin\;B}=\frac{c}{sin\;C} =\rm k\)
Calculations:
Consider, a, b, c are the sides of triangle and A, B, C are the angles of the triangle.
We know that, \(\rm \frac{a}{\ sin\;A}=\frac{b}{sin\;B}=\frac{c}{sin\;C} =\rm k\)
⇒\(\rm \ sin \;A = \frac{a}{k}, \ sin\;B = \frac{b}{k}, \ sin\;C = {c}{k}\)
Given , sin A + sin B = sin C
⇒ A + B = C.
⇒ a + b = c
This is not posible.
Hence, there exists no triangle ABC for which sin A + sin B = sin C.
(2) Given, the angles of a triangle are in the ratio 1 : 2 : 3.
Consider the angles of a triangle are A = x, B = 2x and C = 3x.
We know that, Sum of the angles of the triangle is 180°
Hence, x + 2x + 3x = 180
6x = 180°
x = 30º
Angles are A = 30º, B = 60º and C = 90º
We know that, \(\rm \frac{a}{\ sin\;A}=\frac{b}{sin\;B}=\frac{c}{sin\;C} =\rm k\)
\(\rm \frac{a}{\ sin\;30}=\frac{b}{sin\;60}=\frac{c}{sin\;90} =\rm k\)
\(\rm \frac{a}{\frac{1}{2}}=\frac{b}{\frac{\sqrt{3}}{2}}=\frac{c}{1} =\rm k\)
\(\rm \frac{a}{1}=\frac{b}{\sqrt{3}}=\frac{c}{2} =\rm k\)
Hence, the angles of a triangle are in the ratio 1 : 2 : 3, then its sides will be in the ratio 1 : √3 : 2..