Question

How do you simplify \[\sin {175^ \circ }\cos {55^ \circ } - \cos {175^ \circ }\sin {55^ \circ }\]?

Answer 1

Hint: The expanded form of \[\sin \left( {a - b} \right)\] is \[\sin \left( a \right)\cos \left( b \right) - \sin \left( b \right)\cos \left( a \right)\]
This expression resembles the expression given in the question. Our aim is to try to fit the given expression in this standard form.

Complete step-by-step solution:
We know that,
\[\sin \left( {a - b} \right) = \sin \left( a \right)\cos \left( b \right) - \sin \left( b \right)\cos \left( a \right)\]
The given expression is indeed the expanded form of the above mentioned identity.
Here,\[a = {175^ \circ }\] and \[b = {55^ \circ }\]
Therefore, substituting \[a = {175^ \circ }\]and \[b = {55^ \circ }\] in the identity \[\sin \left( {a - b} \right)\] we will get
\[\sin {175^ \circ }\cos {55^ \circ } - \cos {175^ \circ }\sin {55^ \circ }\]
\[ \Rightarrow \sin \left( {{{175}^ \circ } - {{55}^ \circ }} \right)\]
\[ \Rightarrow \sin \left( {{{120}^ \circ }} \right)\]
Now, we have to find the value of \[\sin \left( {{{120}^ \circ }} \right)\]
To find the value of \[\sin \left( {{{120}^ \circ }} \right)\], we will try to make it in terms of some other standard values like \[{180^ \circ }\] ,\[{60^ \circ }\]etc.
We know that, \[120^\circ = 180^\circ - 60^\circ \]
Therefore, we can write \[\sin \left( {{{120}^ \circ }} \right)\] as \[\sin \left( {{{180}^ \circ } - 60} \right)\]
Also,
\[\sin \left( {{{180}^ \circ } - x} \right) = \sin \left( x \right)\]
Hence, \[\sin \left( {{{180}^ \circ } - 60} \right)\] will become \[\sin \left( {{{60}^ \circ }} \right)\]

Hence,\[\sin {175^ \circ }\cos {55^ \circ } - \cos {175^ \circ }\sin {55^ \circ }\] =\[\sin \left( {{{60}^ \circ }} \right)\]

Note: Sine or the sine function is one of the three primary functions in trigonometry, the others being cosine, and tan functions. The sine x or sine theta can be defined as the ratio of the opposite side of a right triangle to its hypotenuse
Cos function (or cosine function) in a triangle is the ratio of the adjacent side to that of the hypotenuse.
Secant is the ratio between the hypotenuses to the shorter side adjacent to an acute angle in a right triangle.
Sin$(A + B) = \operatorname{Sin} A\cos B + \operatorname{Cos} A + \operatorname{Sin} B$
To find the value of\[\sin \left( {{{120}^ \circ }} \right)\], we will use the addition formula and values of these angles.
\[\sin \left( {{{120}^ \circ }} \right)\]= \[\sin \left( {90 + 30} \right)\]
Now using the formula,
\[\sin \left( {a + b} \right) = \sin \left( a \right)\cos \left( b \right) + \sin \left( b \right)\cos \left( a \right)\]
We can write;
 \[\sin \left( {{{120}^ \circ }} \right) = \] \[\sin \left( {a + b} \right) = \sin \left( {90} \right)\cos \left( {30} \right) + \sin \left( {30} \right)\cos \left( {90} \right)\]
Now putting the values \[\sin \left( {{{90}^ \circ }} \right)\], \[\sin \left( {{{30}^ \circ }} \right)\], \[\cos \left( {{{90}^ \circ }} \right)\]and \[\cos \left( {{{30}^ \circ }} \right)\] from the table above, we get;
\[\sin 120 = \left( 1 \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right) - \left( 0 \right)\left( {\dfrac{1}{2}} \right)\]
\[\sin 120 = \left( {\dfrac{{\sqrt 3 }}{2}} \right)\]
This is the standard way of finding the numerical values of the trigonometric ratios. But it is indeed necessary to know some of the standard values before approaching these.