Question

Find‌ ‌the‌ ‌value‌ ‌of‌ ‌the expression $\cos‌ ‌{{24}^{\circ‌ ‌}}+\cos‌ ‌{{55}^{\circ‌ ‌}}+\cos‌ ‌{{125}^{\circ‌ ‌}}+\cos‌{{204}^{\circ‌ ‌}}$‌ ‌

Answer 1

We solve this problem by converting all the given angles in the interval of $\left[ {{0}^{\circ }},{{90}^{\circ }} \right]$ by using the standard conversions of the cosine ratio. We use the standard conversions of cosine ratio as,
(1) $\cos \left( {{180}^{\circ }}-\theta \right)=-\cos \theta$
(2) $\cos \left( {{180}^{\circ }}+\theta \right)=-\cos \theta$
By using these conversions we convert the required angles and find the required answer.

Complete step-by-step solution:
We are asked to find the value of $\cos {{24}^{\circ }}+\cos {{55}^{\circ }}+\cos {{125}^{\circ }}+\cos {{204}^{\circ }}$
Let us assume that the required value as,
$\Rightarrow x=\cos {{24}^{\circ }}+\cos {{55}^{\circ }}+\cos {{125}^{\circ }}+\cos {{204}^{\circ }}............(i)$
Here, we can see that first two angles ${{24}^{\circ }},{{55}^{\circ }}$ are in the interval $\left[ {{0}^{\circ }},{{90}^{\circ }} \right]$
Now, let us convert the angle ${{125}^{\circ }}$ in the interval $\left[ {{0}^{\circ }},{{90}^{\circ }} \right]$
Here, we can see that the angle ${{125}^{\circ }}$ can be written as difference of ${{180}^{\circ }}$ and ${{55}^{\circ }}$ that is,
$\Rightarrow \cos {{125}^{\circ }}=\cos \left( {{180}^{\circ }}-{{55}^{\circ }} \right)$
We know that the standard conversion that is $\cos \left( {{180}^{\circ }}-\theta \right)=-\cos \theta$
By using this conversion in above equation then we get,
$\Rightarrow \cos {{125}^{\circ }}=-\cos {{55}^{\circ }}$
Now, let us convert the angle ${{204}^{\circ }}$ in the interval $\left[ {{0}^{\circ }},{{90}^{\circ }} \right]$
Here, we can see that the angle ${{204}^{\circ }}$ can be written as sum of ${{180}^{\circ }}$ and ${{24}^{\circ }}$ that is,
$\Rightarrow \cos {{204}^{\circ }}=\cos \left( {{180}^{\circ }}+{{24}^{\circ }} \right)$
We know that the standard conversion that is $\cos \left( {{180}^{\circ }}+\theta \right)=-\cos \theta$
By using this conversion in above equation then we get,
$\Rightarrow \cos {{204}^{\circ }}=-\cos {{24}^{\circ }}$
Now, let us take the equation (i) and replace the required values then we get,
$\begin{aligned} & \Rightarrow x=\cos {{24}^{\circ }}+\cos {{55}^{\circ }}+\left( -\cos {{55}^{\circ }} \right)+\left( -\cos {{24}^{\circ }} \right) \\\ & \Rightarrow x=0 \\\ \end{aligned}$
Therefore, we can conclude that the required value of $\cos {{24}^{\circ }}+\cos {{55}^{\circ }}+\cos {{125}^{\circ }}+\cos {{204}^{\circ }}$ is ‘0’ that is,
$\therefore \cos {{24}^{\circ }}+\cos {{55}^{\circ }}+\cos {{125}^{\circ }}+\cos {{204}^{\circ }}=0$

Note: We need to note that all the terms are in cosine ratio so that the required conversion should also be done in cosine ratio only.
We know that we can convert the cosine ratio to sine ratio and vice versa. But in that process we get two different ratios in same equation which will be difficult to find the result as all the given angles ${{24}^{\circ }},{{55}^{\circ }},{{125}^{\circ }},{{204}^{\circ }}$ do not have the values in standard trigonometric table.
So we need to convert all angles in either sine or cosine ratio. As we can see that the first two terms need not to be converted and are in cosine ratio, we need to convert remaining also in cosine ratio for better result.