SATHEE: Trigonometrical Ratios and Identities 1 Question 3

Trigonometrical Ratios and Identities 1 Question 3

3. If the lengths of the sides of a triangle are in $A P$ and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is (2019 Main, 8 April II)

(a) $3 : 4 : 5$

(b) $4 : 5 : 6$

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Answer:

Correct Answer: 3. (b)

Solution:

Let $a, b$ and $c$ be the lengths of sides of a $△ A B C$ such that $a < b < c$ .

Since, sides are in AP.

$∴ 2 b = a + c$

Then, $∠ C = 2 θ$ [according to the question]

So, $∠ B = π - 3 θ$

On applying sine rule in Eq. (i), we get

$2 \sin B = \sin A + \sin C$

$\Rightarrow 2 \sin (π - 3 θ) = \sin θ + \sin 2 θ$ [from Eq. (ii)]

$\Rightarrow 2 \sin 3 θ = \sin θ + \sin 2 θ$

$\Rightarrow 2 [3 \sin θ - 4 \sin^{3} θ] = \sin θ + 2 \sin θ \cos θ$

$\Rightarrow 6 - 8 \sin^{2} θ = 1 + 2 \cos θ [∵ \sin θ$ can not be zero $]$

$\Rightarrow 6 - 8 (1 - \cos^{2} θ) = 1 + 2 \cos θ$

$\Rightarrow 8 \cos^{2} θ - 2 \cos θ - 3 = 0$

$\Rightarrow (2 \cos θ + 1) (4 \cos θ - 3) = 0$

$\Rightarrow \cos θ = \frac{3}{4}$

$orcos θ = - \frac{1}{2}$ (rejected).

Clearly, the ratio of sides is $a : b : c$

$\begin{aligned} = \sin θ : \sin 3 θ : \sin 2 θ \\ = \sin θ : (3 \sin θ - 4 \sin^{3} θ) : 2 \sin θ \cos θ \\ = 1 : (3 - 4 \sin^{2} θ) : 2 \cos θ \\ = 1 : (4 \cos^{2} θ - 1) : 2 \cos θ \\ = 1 : \frac{5}{4} : \frac{6}{4} = 4 : 5 : 6 \end{aligned}$

Trigonometrical Ratios and Identities 1 Question 3

3. If the lengths of the sides of a triangle are in $A P$ and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is (2019 Main, 8 April II)

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Trigonometrical Ratios and Identities 1 Question 3

3. If the lengths of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is (2019 Main, 8 April II)

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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3. If the lengths of the sides of a triangle are in $A P$ and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is (2019 Main, 8 April II)