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Mathematics

Assertion (A): In adjoining triangle ABC, sinA cosA = 1225\dfrac{12}{25}.

Reason (R): cos A = 1sin A\dfrac{1}{\text{sin A}}.

In adjoining triangle ABC, sinA cosA =: Trigonometrical Ratios, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

  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

In triangle ABC,

Using pythagoras theorem,

⇒ AC2 = AB2 + BC2

⇒ AC2 = 42 + 32

⇒ AC2 = 16 + 9

⇒ AC2 = 25

⇒ AC = 25\sqrt{25}

⇒ AC = 5

By formula,

sin A=PerpendicularHypotenuse=BCAC=35.\text{sin A} = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}}\\[1em] = \dfrac{BC}{AC}\\[1em] = \dfrac{3}{5}.

By formula,

cos A=BaseHypotenuse=ABAC=45.\text{cos A} = \dfrac{\text{Base}}{\text{Hypotenuse}}\\[1em] = \dfrac{AB}{AC}\\[1em] = \dfrac{4}{5}.

Substituting values we get,

sin A cos A = 35×45=3×45×5=1225\dfrac{3}{5} \times \dfrac{4}{5} = \dfrac{3 \times 4}{5 \times 5} = \dfrac{12}{25}.

∴ Assertion (A) is true.

By formula,

1sin A\dfrac{1}{\text{sin A}} = cosec A ≠ cos A

∴ Reason (R) is false.

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

Hence, option 1 is the correct option.

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