Assertion (A): The value of cosine decreases as the angle increases from 0^\circto {90}^\circ. Reason (R): Cosine is defined as the ratio of the adjacent side to the angle under consideration and the hypotenuse, and as the angle increases

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Solution

Q.
Assertion (A): The value of cosine decreases as the angle increases from $0 ° t o 90 °$

Reason (R): Cosine is defined as the ratio of the adjacent side to the angle under consideration and the hypotenuse, and as the angle increases, the length of the adjacent side decreases relative to the constant hypotenuse

1 Both A and R are true, and R is the correct explanation of A

2 Both A and R are true, but R is not the correct explanation of A

3 A is true, but R is false

4 A is false, but R is true

Ans.
(1)

$I n △ A B C r i g h t a n g l e d a t B, l e t ∠ C = θ ∴ \cos θ = B C / A C = b / h$

[where b stands for base (adjacent side) and h stands for hypotenuse]

Obviously, by keeping hypotenuse constant

As $θ$ increases ⇒ b decreases

$\Rightarrow b / h d e c r e a s e s \Rightarrow \cos θ d e c r e a s e s .$

Hence, Assertion (A) is true. Also Reason (R) represents the same facts and supports the Assertion (A). Therefore, Reason (R) is true and correctly explains the Assertion (A).

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