If A B C D is a cyclic quadrilateral with A B = 6 ,   B C = 4 ,   C D = 5 ,   D A = 3 and ∠ A B C = θ , then cos θ =

Question

If A B C D is a cyclic quadrilateral with A B = 6 , &#160; B C = 4 , &#160; C D = 5 , &#160; D A = 3 and &#8736; A B C = &#952; , then cos &#952; =

Aicharya · Accepted Answer

The figure below represents the quadrilateral.

In triangle $A B C$ ,

$A C^{2} = A B^{2} + B C^{2} - 2 (A B) (A C) \cos θ$

$\Rightarrow A C^{2} = 6^{2} + 4^{2} - 2 \times 6 \times 4 \times \cos θ$

$\Rightarrow A C^{2} = 52 - 48 \cos θ . . . (i)$

Similarly, in triangle $A D C,$

$A C^{2} = A D^{2} + D C^{2} - 2 \times A D \times D C \cos (180 ° - θ)$

$\Rightarrow A C^{2} = 3^{2} + 5^{2} + 2 \times 3 \times 5 \times \cos θ$

$\Rightarrow A C^{2} = 34 + 30 \cos θ . . . (ii)$

From equations $(i) & (ii)$ ,

$52 - 48 \cos θ = 34 + 30 \cos θ$

So, $\cos θ = \frac{3}{13}$

If A B C D is a cyclic quadrilateral with A B = 6 ,   B C = 4 ,   C D = 5 ,   D A = 3 and…