Sine Rule MCQ Quiz - Objective Question with Answer for Sine Rule - Download Free PDF

Explanation:

We are given the triangle with angles:

Step 1: Check if the sum of the angles is 180°:

This confirms that the angles satisfy the angle sum property of a triangle.

Statement I. The triangle is right-angled.

Since, the triangle is right-angled.

Statement III: III. The angles A, C and B of the triangle are in AP.

The angles are in Arithmetic Progression because:

This confirms that the angles are in AP.

Statement II is not correct because there is no mention of a side being 3 times the other.

∴ The correct answer is Option (I) and (III) are correct.

Hence, the correct answer is Option 3.

Calculation:

We are given the equation for the ratio of sides using the Sine Rule:

Step 3: Use the identity for sin(3x), which is , and substitute it into the equation:

We have two possible solutions for this equation:

, which gives x = (not valid in this case).

, which simplifies to:

∴ The correct answer is Option (2)

Calculation

Let the sides be .

It is understood that d > 0\) and from the figure, is greatest and is smallest.

By given condition, .

And hence .

Hence by sine rule we have,

or .

and .

.

.

or .

sides are

or .

Hence the required ratio is .

Then the sum of ratio of its sides is  = 15

We know that and .

Thus, .

Using above relation we can claim that is a right angled triangle with AB as hypotenuse, right angled at C.

Calculation

(Sine Rule)

From sine rule

Using equation

But maximum value of

OR

is right angled triangle

Hence option 3 is correct

Given:

 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.

Concepts:

Sine law ⇔ 

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,

⇒ sin A = (√3/2) × (1/√3)

⇒ sin A = 1/2

⇒ A = 30° 

Hence, option (1) is correct. 

Concept:

Sine Rule:

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

⇒ 

⇒ sin B = 

⇒ B = 30∘ = 

Additional Information

Cosine Rule:

cos A = 

cos B = 

cos C = 

Concept:

In ΔABC, sine rule

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.

Hence,  ΔABC is right angled triangle.

....(1)

By sine rule, we have

⇒

⇒

⇒ and 

 Equation (1) becomes, 

⇒ ....(2)

Since, and sum of all angles of triangle is 180º

⇒Max () = 

Equation (2) becomes,

Concept:

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:

sin A - cos B = 

Calculation:

1. We have,

sin 

Hwence, Right angle triangle can be drawn as,

Here, P = 1
H = 3
B = x

Using Pythagoras theorem,

3² = 1² + x²

x² = 8

Now , Cosec C = 

Hence Ststement (1) is not correct.

2. Given,

b cos B = c cos C

Using the sine rule,

⇒ 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 × sin

⇒ 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.

Concept:

In triangle ABC, According to Sine rule

 Where, a, b, c are sides and A, B, C are angles of triangle.

Calculation:

In a triangle ABC, c = 2, A = 45°, ,

Using sine rule in given triangle,

Hence, option (1) is correct.

Concept:

Concept:

In triangle ABC, According to the Sine rule

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

⇒ 

⇒ 

⇒ 

⇒ b = √2 

Concept:

Consider a triangle ABC,

Cosine rule:

Sine rule:

Calculation:

Given: a = (1 + √ 3) cm, b = 2 cm and ∠C = 60°

We know that,

⇒ 2(1 + √3) = 1 + 3 + 2√3 + 4 – c²

⇒ c² = 6

⇒ c = √6

Using sin rule we get,

⇒ sin B = (2 / √6) × sin 60°

⇒ 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°.

Concept:

Sine Rule:

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

sin B = 1

B = 90∘ = π/2

Additional Information

Cosine Rule:

cos A = 

cos B = 

cos C = 

Concept:

Calculations:

Consider, a, b, c are the sides of triangle and A, B, C are the angles of the triangle. 

We know that, 

⇒

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, 

Hence, the angles of a triangle are in the ratio 1 : 2 : 3, then its sides will be in the ratio 1 : √3 : 2..