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How do you find the value of \[\cos 105^{ \circ}\] without using the calculator?

Answer
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Hint: We are given a trigonometric angle and we have to find its numerical value without using the calculator as a value of the angle is not in the trigonometric table chart so first we will convert the value of one angle into the some of those two angles whose values can be accessed from the chart then we will use the formula.
cos(A+B)= cosAcosB – sinAsinB
Here A and B are the two component numbers in which the angle is broken as a sum of two numbers whose value can be found from the normal trigonometry angle table then we will put the value of these angles from the table and then by solving the expression will find the value of the expression. Then we'll put the value of the angles from the table and then by solving the expression we will find the value of the expression.

Complete step-by-step answer:
Step1: We are given the trigonometric angle that is \[\cos 105\] we have found its numerical value without using the calculator. Value of 105 is not in the trigonometric angle table so we will break 105 as a sum of two angles whose values are given in the table.
Here 105 $= {45^ \circ + 60^ \circ}$
So we can write it as,
$ \Rightarrow \cos \left( {45^ \circ + 60^ \circ} \right)$
Step2: Now we will use the formula of cos(A+B)= cosAcosB – sinAsinB
Here A$ = 45^ \circ$ and B$ = 60^ \circ$
Substituting the values in the formula we will get:
The value of $45^ \circ $ and $60^ \circ$ are given in the table so we will put the value from the table into the expression.
$\cos 45^ \circ = \dfrac{1}{{\sqrt 2 }};\cos 60 ^ \circ= \dfrac{1}{2};\sin 45^ \circ = \dfrac{1}{{\sqrt 2 }};\sin 60^ \circ = \dfrac{{\sqrt 3 }}{2}$
On substituting the values in the expression we will get:
$ \Rightarrow \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{1}{2}} \right) - \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right)$
On solving we will get:
$ \Rightarrow \dfrac{1}{{2\sqrt 2 }} - \dfrac{{\sqrt 3 }}{{2\sqrt 2 }}$
$ \Rightarrow \dfrac{{1 - \sqrt 3 }}{{2\sqrt 2 }}$
Step3: On further rationalizing the denominator we will multiply by $\sqrt 2 $ in both numerator and denominator.
$ \Rightarrow \dfrac{{1 - \sqrt 3 }}{{2\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}$
$ \Rightarrow \dfrac{{\sqrt 2 - \sqrt 6 }}{4}$

Hence the value is $\dfrac{{\sqrt 2 - \sqrt 6 }}{4}$

Note:
 In such types of questions students mainly did not get an approach how to solve such questions. Students should keep in mind that to find the value of an angle of digits whose values cannot be found graphically or by table then just split that number into the sum or difference of two such angles whose values can be found by the table. Be careful while applying the formula as there are 4 formulas of this kind. Students mainly mix the formulas and also apply it wrong. And revise the values from the table before solving the question as students get confused in the values of the angle as the values are sometimes the same for different angles. By following this question will get solved easily.