Assertion (A): {sin{\ }}^2\theta\le1\mathrm{and}{cos}^\ 2??1 Reason (R): Hypotenuse is the longest side of

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Q.
$A s s e r t i o n (A) : {s i n}^{2} θ \leq 1 and {c o s}^{} 2 θ \leq 1$

$R e a s o n (R) : H y p o t e n u s e i s t h e l o n g e s t s i d e o f t h e t r i a n g l e .$

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)

Hypotenuse is the longest side of the triangle.
$∴ s i n θ = \frac{Opposite side}{Hypotenuse} < 1 and c o s θ = \frac{Adjacent side}{Hypotenuse} < 1 a n d h e n c e {s i n}^{2} θ \leq 1 and {c o s}^{} 2 θ \leq 1 .$

Thus, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Solution

Q.
$A s s e r t i o n (A) : {s i n}^{2} θ \leq 1 and {c o s}^{} 2 θ \leq 1$

$R e a s o n (R) : H y p o t e n u s e i s t h e l o n g e s t s i d e o f t h e t r i a n g l e .$

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.

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Solution

Q. Assertion(A): sin 2θ≤1 and cos 2θ≤1 Reason (R): Hypotenuse is the longest side of the triangle.

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) Hypotenuse is the longest side of the triangle. ∴sinθ=Opposite sideHypotenuse<1andcosθ=Adjacent sideHypotenuse<1and hence sin 2θ≤1 and cos 2θ≤1. Thus, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

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Q.
$A s s e r t i o n (A) : {s i n}^{2} θ \leq 1 and {c o s}^{} 2 θ \leq 1$

$R e a s o n (R) : H y p o t e n u s e i s t h e l o n g e s t s i d e o f t h e t r i a n g l e .$