Question

Given that \[\cos {{50}^{\circ }}{{18}^{'}}=0.6388\] and \[\cos {{50}^{\circ }}{{42}^{'}}=0.6334,\] then the possible value of \[\cos {{50}^{\circ }}{{20}^{'}}\] is:
(a) 0.6293
(b) 0.6307
(c) 0.6361
(d) 0.6414

Answer 1

Hint: We are given the value of cos at two angles. We will first find the difference in their angle and then find the difference of the value of cos at these angles. We get a relation that 24’ at an angle gives 0.0054 difference. In the value of cos using the unitary method, we evaluate the difference for 2’ of angle, then we subtract that the difference from the value of \[\cos {{50}^{\circ }}{{18}^{'}}.\]

Complete step-by-step answer:
We are given that the value of cos at \[\cos {{50}^{\circ }}{{18}^{'}}\] is 0.6388.
\[\cos {{50}^{\circ }}{{18}^{'}}=0.6388\]
Also, we have that cos at \[\cos {{50}^{\circ }}{{42}^{'}}\] is 0.6334.
\[\cos {{50}^{\circ }}{{42}^{'}}=0.6334\]
We will now calculate the difference of the angle. So, the difference of \[{{50}^{\circ }}{{18}^{'}}\] and \[{{50}^{\circ }}{{42}^{'}}\] is
\[{{50}^{\circ }}{{42}^{'}}-{{50}^{\circ }}{{18}^{'}}={{24}^{'}}\]
Now we will look for the difference in the value of cos of these angles.
\[{{50}^{\circ }}{{42}^{'}}-{{50}^{\circ }}{{18}^{'}}\]
\[\Rightarrow 0.6388-0.6334=0.0054\]
We get the difference in the value of cos of angles as 0.0054.
Hence we have got that for 24’ difference of angle, cos differ by 0.0054.
\[\Rightarrow {{24}^{'}}=0.0054\]
Now using the unitary method, we get,
\[{{1}^{'}}=\dfrac{0.0054}{24}\]
\[\Rightarrow {{1}^{'}}=0.000225\]
As \[\cos {{50}^{\circ }}{{18}^{'}}\] and \[\cos {{50}^{\circ }}{{20}^{'}}\] differ by 2’. So, we get,
\[{{2}^{'}}=0.000225\times 2\]
\[\Rightarrow {{2}^{'}}=0.00045\]
\[\Rightarrow {{2}^{'}}\cong 0.0005\]
Now we subtract this value from \[\cos {{50}^{\circ }}{{18}^{'}}\] to get the required solution.
\[\cos {{50}^{\circ }}{{20}^{'}}=\cos {{50}^{\circ }}{{18}^{'}}-0.0005\]
\[\Rightarrow \cos {{50}^{\circ }}{{20}^{'}}=0.6388-0.0005\]
\[\Rightarrow \cos {{50}^{\circ }}{{20}^{'}}=0.6383\]
So, we get \[\cos {{50}^{\circ }}{{20}^{'}}\] as 0.6383.
We check from the options and see that the nearest value is 0.6361.
Hence, the option (c) is the right answer.
Note: We can also use an alternate method. We know that from 0 to \[{{90}^{\circ }}\] the value of cos decreases with an increase of angle as \[\cos {{50}^{\circ }}{{20}^{'}}\] lie between \[\cos {{50}^{\circ }}{{18}^{'}}\] and \[\cos {{50}^{\circ }}{{42}^{'}}.\] So, the value of \[\cos {{50}^{\circ }}{{20}^{'}}\] will lie between these values only. So, options (a), (b) and (d) are all values that are either greater than or less than the value of \[\cos {{50}^{\circ }}{{18}^{'}}\] and \[\cos {{50}^{\circ }}{{42}^{'}}.\] So, the right option is (c). 0.6361.