3. Value of (sin 60° cos 30° + sin 30° cos 60° -tan 45º ) is-a) 2

1.	3. Value of (sin 60° cos 30° + sin 30° cos 60° -tan 45º ) is-a) 2.b) 4c) 1d)none of these
Answer» $\rm \implies sin \: 60 \degree \: cos \: 30\degree + sin \: 30 \degree \: cos \: 60\degree - tan \: 45\degree$ We know that :- Sin 60° = √3/2 Cos 30° = √3/2 Sin 30° = 1/2 cos 30° = 1/2 tan 45° = 1 $\rm \implies \bigg( \dfrac{ \sqrt{3} }{2} \times\dfrac{ \sqrt{3} }{2} \bigg )+ \bigg( \dfrac{1}{2} \times \dfrac{1}{2} \bigg ) -1$ $\rm \implies \bigg( \dfrac{{3} }{4} \bigg )+ \bigg( \dfrac{1}{4} \bigg ) -1$ $\rm \implies \dfrac{{3} }{4} + \dfrac{1}{4} -1$ LCM of 4,4 and 1 = 4 $\rm \implies \dfrac{{3} + 1 - 4 }{4}$ $\rm \implies \dfrac{4 - 4 }{4} = 0$ Therefore :- $\rm \implies sin \: 60 \degree \: cos \: 30\degree + sin \: 30 \degree \: cos \: 60\degree - tan \: 45\degree = 0$ ____________________ TRIGONOMETRIC table :- $\begin{array}{ \|c \|c\|c\|c\|c\|c\|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3} }{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }& 1 & \sqrt{3} & \rm Not \: De fined \\ \\ \rm cosec A & \rm Not \: De fined & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm Not \: De fined \\ \\ \rm cot A & \rm Not \: De fined & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}$

3. Value of (sin 60° cos 30° + sin 30° cos 60° -tan 45º ) is-a) 2.b) 4c) 1d)none of these

Answer»

$\rm \implies sin \: 60 \degree \: cos \: 30\degree + sin \: 30 \degree \: cos \: 60\degree - tan \: 45\degree$

We know that :-

Sin 60° = √3/2

Cos 30° = √3/2

Sin 30° = 1/2

cos 30° = 1/2

tan 45° = 1

$\rm \implies \bigg( \dfrac{ \sqrt{3} }{2} \times\dfrac{ \sqrt{3} }{2} \bigg )+ \bigg( \dfrac{1}{2} \times \dfrac{1}{2} \bigg ) -1$

$\rm \implies \bigg( \dfrac{{3} }{4} \bigg )+ \bigg( \dfrac{1}{4} \bigg ) -1$

$\rm \implies \dfrac{{3} }{4} + \dfrac{1}{4} -1$

LCM of 4,4 and 1 = 4

$\rm \implies \dfrac{{3} + 1 - 4 }{4}$

$\rm \implies \dfrac{4 - 4 }{4} = 0$

Therefore :-

$\rm \implies sin \: 60 \degree \: cos \: 30\degree + sin \: 30 \degree \: cos \: 60\degree - tan \: 45\degree = 0$

____________________

TRIGONOMETRIC table :-

$\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3} }{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }& 1 & \sqrt{3} & \rm Not \: De fined \\ \\ \rm cosec A & \rm Not \: De fined & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm Not \: De fined \\ \\ \rm cot A & \rm Not \: De fined & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}$