Answer
Verified
474.3k+ views
Hint: Substitute the values of $\sin 30{}^\circ ,\sin 60{}^\circ ,\cos 30{}^\circ ,\cos 60{}^\circ $ and simplify and hence find the value of the given expression
Complete step-by-step answer:
To solve the given question, we need to remember the values of $\sin 30{}^\circ ,\sin 60{}^\circ ,\cos 30{}^\circ $ and $\cos 60{}^\circ $
Consider the following tables of values of sine, cosine, tangent, cotangent, secant and cosecant for angles of measure $0{}^\circ ,30{}^\circ ,45{}^\circ ,60{}^\circ ,90{}^\circ $
From the above table, we have
$\sin 30{}^\circ =\dfrac{1}{2},\sin 60{}^\circ =\dfrac{\sqrt{3}}{2},\cos 30{}^\circ =\dfrac{\sqrt{3}}{2},\cos 60{}^\circ =\dfrac{1}{2}$
Substituting the values of $\sin 30{}^\circ ,\sin 60{}^\circ ,\cos 30{}^\circ ,\cos 60{}^\circ $, we get
$S=\dfrac{1}{2}\times \dfrac{1}{2}+\dfrac{\sqrt{3}}{2}\times \dfrac{\sqrt{3}}{2}$
Simplifying, we get
$S=\dfrac{1}{4}+\dfrac{3}{4}=\dfrac{4}{4}=1$
Hence, we have
$\sin 30{}^\circ \cos 60{}^\circ +\sin 60{}^\circ \cos 30{}^\circ $ is equal to 1.
Therefore the correct answer is option (c).
Note: [1] Alternatively, you can use the fact that $\cos \left( 90{}^\circ -\theta \right)=\sin \theta $ and $\sin \left( 90{}^\circ -\theta \right)=\cos \theta $ and then write $\cos 60{}^\circ $ as $\cos \left( 90{}^\circ -30{}^\circ \right)=\sin 30{}^\circ $ and $\sin 60{}^\circ $ as $\sin \left( 90{}^\circ -30{}^\circ \right)=\cos 30{}^\circ $
Hence the expression becomes
$S={{\sin }^{2}}30{}^\circ +{{\cos }^{2}}30{}^\circ $
Now, we know that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$
Hence, we have
$S=1$, which is the same as obtained above.
Hence option [c] is the correct answer.
[2] Alternative
We know that $\sin A\cos B+\cos A\sin B=\sin \left( A+B \right)$
Put $A=30{}^\circ $ and $B=60{}^\circ $.
Hence, we have
$\sin \left( 30{}^\circ +60{}^\circ \right)=\sin 30{}^\circ \cos 60{}^\circ +\cos 30{}^\circ \sin 60{}^\circ =S$
Hence, we have
$S=\sin 90{}^\circ $
From the above table, we have $\sin 90{}^\circ =1$
Hence, we have $S=1$, which is the same as obtained above.
Hence option [c] is the correct answer.
Complete step-by-step answer:
To solve the given question, we need to remember the values of $\sin 30{}^\circ ,\sin 60{}^\circ ,\cos 30{}^\circ $ and $\cos 60{}^\circ $
Consider the following tables of values of sine, cosine, tangent, cotangent, secant and cosecant for angles of measure $0{}^\circ ,30{}^\circ ,45{}^\circ ,60{}^\circ ,90{}^\circ $
From the above table, we have
$\sin 30{}^\circ =\dfrac{1}{2},\sin 60{}^\circ =\dfrac{\sqrt{3}}{2},\cos 30{}^\circ =\dfrac{\sqrt{3}}{2},\cos 60{}^\circ =\dfrac{1}{2}$
Substituting the values of $\sin 30{}^\circ ,\sin 60{}^\circ ,\cos 30{}^\circ ,\cos 60{}^\circ $, we get
$S=\dfrac{1}{2}\times \dfrac{1}{2}+\dfrac{\sqrt{3}}{2}\times \dfrac{\sqrt{3}}{2}$
Simplifying, we get
$S=\dfrac{1}{4}+\dfrac{3}{4}=\dfrac{4}{4}=1$
Hence, we have
$\sin 30{}^\circ \cos 60{}^\circ +\sin 60{}^\circ \cos 30{}^\circ $ is equal to 1.
Therefore the correct answer is option (c).
Note: [1] Alternatively, you can use the fact that $\cos \left( 90{}^\circ -\theta \right)=\sin \theta $ and $\sin \left( 90{}^\circ -\theta \right)=\cos \theta $ and then write $\cos 60{}^\circ $ as $\cos \left( 90{}^\circ -30{}^\circ \right)=\sin 30{}^\circ $ and $\sin 60{}^\circ $ as $\sin \left( 90{}^\circ -30{}^\circ \right)=\cos 30{}^\circ $
Hence the expression becomes
$S={{\sin }^{2}}30{}^\circ +{{\cos }^{2}}30{}^\circ $
Now, we know that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$
Hence, we have
$S=1$, which is the same as obtained above.
Hence option [c] is the correct answer.
[2] Alternative
We know that $\sin A\cos B+\cos A\sin B=\sin \left( A+B \right)$
Put $A=30{}^\circ $ and $B=60{}^\circ $.
Hence, we have
$\sin \left( 30{}^\circ +60{}^\circ \right)=\sin 30{}^\circ \cos 60{}^\circ +\cos 30{}^\circ \sin 60{}^\circ =S$
Hence, we have
$S=\sin 90{}^\circ $
From the above table, we have $\sin 90{}^\circ =1$
Hence, we have $S=1$, which is the same as obtained above.
Hence option [c] is the correct answer.
Recently Updated Pages
What number must be added to each term of the ratio class 11 maths CBSE
What is the nucleon number of an atom A The number class 11 chemistry CBSE
Nucleolus was discovered by A Robert Brown B Leeuwenhoek class 11 biology CBSE
Nucleolus is formed by A Mitochondria B Nucleus and class 11 biology CBSE
How do nuclear charge and shielding affect the periodic class 11 chemistry CBSE
Notice these expressions in the text Infer their meaning class 11 english CBSE
Trending doubts
Find the value of the expression given below sin 30circ class 11 maths CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
In the tincture of iodine which is solute and solv class 11 chemistry CBSE
On which part of the tongue most of the taste gets class 11 biology CBSE
State and prove Bernoullis theorem class 11 physics CBSE
Who is the leader of the Lok Sabha A Chief Minister class 11 social science CBSE