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Mathematics

Assertion (A): If 0° < A + B ≤ 90°, A > B and cos (A + B) = 12\dfrac{1}{2} = sin (A - B), then we can say that A = 45° and B = 15°.

Reason (R): sin 60° = cos 60°.

  1. Assertion (A) is true, Reason (R) is false.

  2. Assertion (A) is false, Reason (R) is true.

  3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

  4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Trigonometrical Ratios

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Answer

Given, cos (A + B) = 12\dfrac{1}{2}

Since 0° < A + B ≤ 90°, the angle whose cosine = 12\dfrac{1}{2} is 60°.

So, A + B = 60° ………………..(1)

sin (A - B) = 12\dfrac{1}{2}

Since A > B, A - B will be positive. The angle whose sine = 12\dfrac{1}{2} is 30°.

So, A - B = 30° ………………..(2)

Adding equations (1) and (2), we get :

⇒ (A + B) + (A - B) = 60° + 30°

⇒ A + B + A - B = 90°

⇒ 2A = 90°

⇒ A = 90°2\dfrac{90°}{2}

⇒ A = 45°

Substituting the value of A in equation (1), we get :

⇒ 45° + B = 60°

⇒ B = 60° - 45°

⇒ B = 15°.

∴ Assertion (A) is true.

sin 60° = 32\dfrac{\sqrt{3}}{2}

cos 60° = 12\dfrac{1}{2}

As sin 60° ≠ cos 60°

∴ Reason (R) is false.

∴ Assertion (A) is true, Reason (R) is false.

Hence, option 1 is the correct option.

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