Question

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

Answer 1

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

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 :-

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

Answer 2

To Find :

The value of -

$\sf{sin60cos30+sin30cos60-tan45}$

Solution:

We Know,

$\bullet\:\sf Trigonometric\:Values :\\\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D{e}fined\end{tabular}}$

So, Simply putting the values,

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

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

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

$\sf{\implies}{\dfrac{4-4}{4}}$

$\sf{\implies 0}$

Therefore, the value of $\sf{sin60cos30+sin30cos60-tan45}$ is 0.

Important points to remember--

$\boxed{\begin{minipage}{6cm} Important Trigonometric identities :- \\ \\ \: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1 \end{minipage}}$

Note:-

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