SATHEE: Properties of Triangles 2 Question 13

Properties of Triangles 2 Question 13

13. If in a $△ A B C, \cos A \cos B + \sin A \sin B \sin C = 1$ , then show that $a : b : c = 1 : 1 : \sqrt{2}$ .

$(1986, 5$ M)

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Solution:

Given, $\cos A \cos B + \sin A \sin B \sin C = 1$

$\begin{array}{lc} \Rightarrow & \sin C = \frac{1 - \cos A \cos B}{\sin A \sin B} \\ \Rightarrow & \frac{1 - \cos A \cos B}{\sin A \sin B} \leq 1 [∵ \sin C \leq 1] \\ \Rightarrow & 1 - \cos A \cos B \leq \sin A \sin B \\ \Rightarrow & \cos (A - B) \geq 1 \\ \Rightarrow & \cos (A - B) = 1 \\ \Rightarrow & A - B = 0 \end{array} [∵ ascos (θ) \leq 1]$

On putting $A = B$ in Eq. (i), we get

$\begin{array}{lc} \sin C = \frac{1 - \cos^{2} A}{\sin^{2} A} \\ \Rightarrow & \sin C = 1 \\ \Rightarrow & C = π / 2 \\ Now, & A + B + C = π \end{array}$

$\begin{aligned} \Rightarrow A + B = \frac{π}{2} \Rightarrow A = \frac{π}{4} ∵ A = B and C = \frac{π}{2} \\ ∴ \sin A : \sin B : \sin C = \sin \frac{π}{4} : \sin \frac{π}{4} : \sin \frac{π}{2} \\ \Rightarrow a : b : c = \frac{1}{\sqrt{2}} : \frac{1}{\sqrt{2}} : 1 \\ = 1 : 1 : \sqrt{2} \end{aligned}$

Properties of Triangles 2 Question 13

13. If in a $△ A B C, \cos A \cos B + \sin A \sin B \sin C = 1$ , then show that $a : b : c = 1 : 1 : \sqrt{2}$ .

Solution:

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Properties of Triangles 2 Question 13

13. If in a △ABC,cos⁡Acos⁡B+sin⁡Asin⁡Bsin⁡C=1, then show that a:b:c=1:1:2.

Solution:

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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13. If in a $△ A B C, \cos A \cos B + \sin A \sin B \sin C = 1$ , then show that $a : b : c = 1 : 1 : \sqrt{2}$ .