Question

Find the value of \[\sin {{50}^{\circ }}\].

Answer 1

Hint: We will first apply the formula \[\cos 3x=4{{\cos }^{3}}x-3\cos x.......(1)\] by putting x equal to 20 to get cos(20) and then we will use \[{{\sin }^{2}}{{20}^{\circ }}+{{\cos }^{2}}{{20}^{\circ }}=1.....(5)\] to find sin(20). Then we will substitute A equal to 30 and B equal to 20 in \[\sin (A+B)=\sin A\cos B+\cos A\sin B\] to get the final answer.

Complete step-by-step answer:

We will use the formula \[\cos 3x=4{{\cos }^{3}}x-3\cos x.......(1)\] and substituting x as 20 in equation (1) we get,

\[\begin{align}

  & \Rightarrow \cos (3\times 20)=4{{\cos }^{3}}(20)-3\cos (20) \\

 & \Rightarrow \cos (60)=4{{\cos }^{3}}(20)-3\cos (20)......(2) \\

\end{align}\]

We know that cos(60) is \[\dfrac{1}{2}\] so substituting it in equation (2) and also putting cos(20) equal to k in equation (2), we get,

\[\Rightarrow \dfrac{1}{2}=4{{k}^{3}}-3k......(3)\]

Rearranging and simplifying equation (3) in a cubic equation we get,

\[\Rightarrow 8{{k}^{3}}-6k-1=0......(4)\]

Solving the cubic equation (4) by hit and trial method we get k equal to 0.94 and other two roots being negative so not taking them into account.

Now we know that cos(20) is equal to 0.94.

Also we know that \[{{\sin }^{2}}{{20}^{\circ }}+{{\cos }^{2}}{{20}^{\circ }}=1.....(5)\]

So putting value of cos(20) in equation (5) we get,

\[{{\sin }^{2}}{{20}^{\circ }}+{{(0.94)}^{2}}=1.....(6)\]

So finding sin(20) by rearranging equation (6) we get,


\[\begin{align}

  & {{\sin }^{2}}{{20}^{\circ }}=1-0.8836 \\

 & {{\sin }^{2}}{{20}^{\circ }}=0.1164 \\

 & \sin {{20}^{\circ }}=\sqrt{0.1164}=0.34......(7) \\

\end{align}\]

Now we can write sin(50) as sin(30+20). So using the formula \[\sin (A+B)=\sin A\cos B+\cos A\sin B\] we get,

\[\sin {{50}^{\circ }}=\sin ({{30}^{\circ }}+{{20}^{\circ }})=\sin {{30}^{\circ }}\cos {{20}^{\circ }}+\cos {{30}^{\circ }}\sin {{20}^{\circ }}.....(8)\]

Now we know the standard angle values and we have also calculated the values of cos(20) and sin(20). So now substituting the value of sin(20) from equation (7) and cos(20) in equation (8) we get,

\[\sin {{50}^{\circ }}=\dfrac{1}{2}\times 0.94+\dfrac{\sqrt{3}}{2}\times 0.34.....(9)\]

So solving equation (9) we get,

\[\sin {{50}^{\circ }}=0.47+0.29=0.76\]

Hence the value of \[\sin {{50}^{\circ }}\] is 0.76.

Note: Remembering the basic trigonometric formulas is the key in solving these types of problems. We may get confused about how to proceed in solving equation (3) but we use a hit and trial method to solve it. In a hurry we can make a mistake in solving equation (9) so we need to be careful while doing this step.