Sine Rule MCQ Quiz in मल्याळम - Objective Question with Answer for Sine Rule - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:

Sine Rule:

In a triangle Δ ABC, 

Where, 

∠A + ∠B + ∠C = π 

Formula used:

1. cos(π - θ) = -cos θ 
2. 1 - sin²θ = cos²θ  
3. cos(A + B) cos(A − B) = cos²A - sin²Bcos(A+B)cos(A−B)=cos2A−sin2B

Calculation:

We have

  (∵ ∠A + ∠B + ∠C = π )

    (∵ cos(π - θ) = -cos θ)

       (∵1 - sin2θ = cos2θ)

From the sine rule,

⇒ sin A = ka, sin B = kb, sin C = kc

Hence, required value

Formula used - 

In a triangle ABC, Sine rule, 

(sinA/a) = (sinB/b) = (sinC/c)

Given - 

∠A = 30°, ∠B = 45°, AC = 3√2 cm

Solution - 

⇒ (sin45°/3√2) = (sin30°/BC)

⇒ (1/6) = {1/(2 BC)}

⇒ BC = 3cm

∴ side BC = 3 cm

Concept:

3 sin α - 4 sin3 α = sin 3α

Explanation:

We know,

3 sin α - 4 sin3 α = sin 3α

Since, α = 

⇒ 3 sin  - 4 sin³  = sin (3× )

⇒ 3 sin α 

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)

are in A.P.

(Given)

Put 0\)

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

Answer : 2

Solution :

In ΔABC, by sine rule,

According to the given condition,

In ΔABC, a = 2 b and

A - B = 60° A = 60° + B

⇒ 

⇒ 

⇒ 2 sin B = sin B cos 60° + cos B sin 60°

⇒  

⇒ 

⇒ 

∴ A = 30° + 60° = 90°

∴ ΔABC is right angled.

Concept Used:

1. Cosine Rule:

2. Half-angle formulas:

Calculation:

Given:

In triangle ABC, c = 4

Expression: 

⇒

⇒

⇒

⇒

⇒

⇒

⇒ (using the cosine rule)

⇒ (since c = 4)

⇒ 16

∴ The value of the expression is 16.

Hence option 2 is correct