Question

Find the value of \[cos{{24}^{\circ }}+cos{{55}^{\circ }}+cos{{125}^{\circ }}+cos{{204}^{\circ }}+cos{{300}^{\circ }}\].
(a) 1
(b) \[\dfrac{1}{2}\]
(c) \[\dfrac{1}{3}\]
(d) \[\dfrac{1}{4}\]

Answer 1

Hint: To solve this we will use certain trigonometric identities such as \[\cos ({{180}^{\circ }}-\theta )=\cos \theta \], \[\cos ({{180}^{\circ }}+\theta )=-\cos \theta \] and\[\cos ({{270}^{\circ }}+\theta )=\sin \theta \]. And by proper substitution and arrangement of the equations we will determine the value of given expression.

Complete step by step solution:
We are given to find the value of\[cos{{24}^{\circ }}+cos{{55}^{\circ }}+cos{{125}^{\circ }}+cos{{204}^{\circ }}+cos{{300}^{\circ }}\].
To calculate the value of this we will use certain trigonometric identities such as \[\cos ({{180}^{\circ }}-\theta )=\cos \theta \],\[\cos ({{180}^{\circ }}+\theta )=-\cos \theta \] and \[\cos ({{270}^{\circ }}+\theta )=\sin \theta \] and apply them using proper substitution.
We have,
\[cos{{24}^{\circ }}+cos{{55}^{\circ }}+cos{{125}^{\circ }}+cos{{204}^{\circ }}+cos{{300}^{\circ }}\]
Splitting \[{{125}^{\circ }}\] as \[{{180}^{\circ }}-{{55}^{\circ }}\], \[{{204}^{\circ }}\] as \[{{180}^{\circ }}+{{24}^{\circ }}\] and \[{{300}^{\circ }}\] as \[{{270}^{\circ }}+{{30}^{\circ }}\] in the above expression we get,
\[\begin{align}
  & cos{{24}^{\circ }}+cos{{55}^{\circ }}+cos{{125}^{\circ }}+cos{{204}^{\circ }}+cos{{300}^{\circ }} \\
 & \Rightarrow cos{{24}^{\circ }}+cos{{55}^{\circ }}+cos{{125}^{\circ }}+cos{{204}^{\circ }}+cos{{300}^{\circ }} \\
 & =cos{{24}^{\circ }}+cos{{55}^{\circ }}+cos\left( {{180}^{\circ }}-{{55}^{\circ }} \right)+cos\left( {{180}^{\circ }}+{{24}^{\circ }} \right)+cos\left( {{270}^{\circ }}+{{30}^{\circ }} \right) \\
\end{align}\]
Now using the trigonometric identity given as \[\cos ({{180}^{\circ }}-\theta )=\cos \theta \] on \[\cos {{125}^{\circ }}\] and\[\cos ({{180}^{\circ }}+\theta )=-\cos \theta \] on \[\cos {{204}^{\circ }}\] in the above expression we get,
\[\Rightarrow cos\text{ }{{24}^{\circ }}+cos{{55}^{\circ }}+cos{{125}^{\circ }}+cos{{204}^{\circ }}+cos{{300}^{\circ }}=cos{{24}^{\circ }}+cos{{55}^{\circ }}-cos{{55}^{\circ }}-cos{{24}^{\circ }}+sin{{30}^{\circ }}\]
Cancelling the repeated terms on the right-hand side of the above expression we get,
\[\Rightarrow cos\text{ }{{24}^{\circ }}+cos{{55}^{\circ }}+cos{{125}^{\circ }}+cos{{204}^{\circ }}+cos{{300}^{\circ }}=cos{{24}^{\circ }}+cos{{55}^{\circ }}-cos{{55}^{\circ }}-cos{{24}^{\circ }}+sin{{30}^{\circ }}\]
Since we know that $sin{{30}^{\circ }}=\dfrac{1}{2}$ , we can substitute it and we will get RHS as
\[\Rightarrow cos\text{ }{{24}^{\circ }}+cos{{55}^{\circ }}+cos{{125}^{\circ }}+\cos {{204}^{\circ }}+\cos {{300}^{\circ }}=\dfrac{1}{2}\]
Therefore, we obtain the value of required expression as \[\dfrac{1}{2}\].
Hence matching the answer from the given options, we get the value of the given expression as \[\dfrac{1}{2}\], which is option (b).
So, option (b) is the correct answer.

Note: The possibility of error in this question can be at the point where we are applying various identities of trigonometric functions to get the required value. Always remember to apply simple and easy to use identities so as to make the substitutions of the expressions easy and understandable.