Trigonometrical Ratios — Assertion-Reason Type Questions

Assertion (A): The value of sin 60° cos 30° + cos 60° sin 30° is 0.

Reason (R): sin 90° = 0 and sin 0° = 1.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Given,

$\Rightarrow \sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ\\[1em] \Rightarrow \dfrac{\sqrt{3}}{2} \times \dfrac{\sqrt{3}}{2} + \dfrac{1}{2} \times \dfrac{1}{2}\\[1em] \Rightarrow \dfrac{3}{4} + \dfrac{1}{4}\\[1em] \Rightarrow \dfrac{4}{4}\\[1em] = 1.$

Assertion (A) is false.

As,

sin 90° = 1 and sin 0° = 0.

So,

Reason (R) is false.

Hence, Option 4 is the correct option.

Assertion (A): In a right angled △ABC, if ∠ABC = 90°, AB = 3 cm, BC = 4 cm, then sin A = cos C.

Reason (R): $\dfrac{\sin θ}{\cos θ}$ = tan θ and sin θ $\times$ cos θ = cot θ.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

In a right angled △ABC,

∠ABC = 90°, AB = 3 cm, BC = 4 cm

By pythagoras theorem,

Hypotenuse² = Perpendicular² + Base²

AC² = (BC)² + (AB)²

AC² = 4² + 3²

AC² = 16 + 9

AC² = 25

AC = $\sqrt{25}$

AC = 5 cm.

$\Rightarrow \sin A = \dfrac{\text{perpendicular}}{\text{hypotenuse}} = \dfrac{BC}{AC} = \dfrac{4}{5}\\[1em] \Rightarrow \cos C = \dfrac{\text{base}}{\text{hypotenuse}} = \dfrac{BC}{AC} = \dfrac{4}{5}\\[1em] \Rightarrow \sin A = \cos C$

Assertion (A) is true.

As, $\dfrac{\sin θ}{\cos θ}$ = tan θ

but sin θ $\times$ cos θ is not equal to cot θ.

So, Reason (R) is false.

Hence, Option 1 is the correct option.

Assertion (A): $\dfrac{\sin 27^\circ}{\cos 63^\circ}$ = 1.

Reason (R): sin(90° - θ) = cos θ and cos(90° - θ) = sin θ.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

$\Rightarrow \dfrac{\sin 27^\circ}{\cos 63^\circ}\\[1em] = \dfrac{\sin (90^\circ - 63^\circ)}{\cos 63^\circ}\\[1em] = \dfrac{\cos 63^\circ}{\cos 63^\circ}\\[1em] = 1$

Assertion (A) is true.

As, sin(90° - θ) = cos θ

and

cos(90° - θ) = sin θ

So,

Reason (R) is true.

Hence, Option 3 is the correct option.