Question

How do you simplify using the half angle formula: - \[\sin \left( {{22}^{\circ }},30' \right)\]?

Answer 1

Hint: First of all convert 30’ into degrees by using the conversion relation: - \[1'=\dfrac{1}{60}\] degrees. Using this conversion, find the total angle, i.e., \[{{22}^{\circ }}30'\] in degrees. Assume this angle as \[\theta \]. Find the value of \[2\theta \] by multiplying \[\theta \] with 2. Now, use the trigonometric identity: - \[{{\sin }^{2}}\theta =\dfrac{1-\cos 2\theta }{2}\] and take square root both the sides to get the answer. Use the value: - \[\cos {{45}^{\circ }}=\dfrac{1}{\sqrt{2}}\].

Complete step by step answer:
Here, we have been provided with the trigonometric function: - \[\sin \left( {{22}^{\circ }},30' \right)\] and we are asked to simplify it using the half angle formula.
Now, first we need to convert the given angle into degrees. The angle is \[{{22}^{\circ }}30'\] which is read as 22 degrees 30 minutes. We know that \[60'={{1}^{\circ }}\], therefore we have,
\[\begin{align}
  & \Rightarrow 1'={{\left( \dfrac{1}{60} \right)}^{\circ }} \\
 & \Rightarrow 30'={{\left( \dfrac{30}{60} \right)}^{\circ }} \\
 & \Rightarrow 30'={{0.5}^{\circ }} \\
\end{align}\]
So, the given angle is \[{{22.5}^{\circ }}\]. Let us assume this angle as \[\theta \]. So, multiplying both the sides with 2, we get,
\[\begin{align}
  & \Rightarrow 2\times \theta =2\times {{22.5}^{\circ }} \\
 & \Rightarrow 2\theta ={{45}^{\circ }} \\
\end{align}\]
Using the trigonometric identity: - \[{{\sin }^{2}}\theta =\dfrac{1-\cos 2\theta }{2}\], we get,
\[\Rightarrow {{\sin }^{2}}\theta =\dfrac{1-\cos 2\theta }{2}\]
Substituting the value of \[2\theta \], we get,
\[\Rightarrow {{\sin }^{2}}\theta =\dfrac{1-\cos {{45}^{\circ }}}{2}\]
Using the value: - \[\cos {{45}^{\circ }}=\dfrac{1}{\sqrt{2}}\], we get,
\[\Rightarrow {{\sin }^{2}}\theta =\dfrac{1-\dfrac{1}{\sqrt{2}}}{2}=\dfrac{\sqrt{2}-1}{2\sqrt{2}}\]
Taking square root both the sides, we get,
\[\Rightarrow \sin \theta ={{\left( \dfrac{\sqrt{2}-1}{2\sqrt{2}} \right)}^{\dfrac{1}{2}}}\]
Rationalizing the denominator by multiplying and dividing with \[\sqrt{2}\], we get,
\[\begin{align}
  & \Rightarrow \sin \theta ={{\left( \dfrac{\sqrt{2}-1}{2\sqrt{2}}\times \dfrac{\sqrt{2}}{\sqrt{2}} \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow \sin \theta ={{\left( \dfrac{2-\sqrt{2}}{4} \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow \sin \theta =\dfrac{{{\left( 2-\sqrt{2} \right)}^{\dfrac{1}{2}}}}{2} \\
 & \Rightarrow \sin \left( {{22}^{\circ }}30' \right)=\dfrac{{{\left( 2-\sqrt{2} \right)}^{\dfrac{1}{2}}}}{2} \\
\end{align}\]
Hence, the value of given trigonometric function is \[\dfrac{{{\left( 2-\sqrt{2} \right)}^{\dfrac{1}{2}}}}{2}\].

Note:
One may note that in trigonometry the term ‘minute’ denotes the angle and not time. Always remember that: - \[60'={{1}^{\circ }}\] and \[60''={{1}^{\circ }}\] where 60’’ represents 60 seconds. It will be very difficult for us to determine the value of \[\sin \left( {{22.5}^{\circ }} \right)\] without using the half angle formula, so you must remember these basic formulas like: - \[{{\cos }^{2}}\theta =\dfrac{1+\cos 2\theta }{2},{{\sin }^{2}}\theta =\dfrac{1-\cos 2\theta }{2}\] and \[\sin 2\theta =2\sin \theta \cos \theta \]. Note that here we have used the second formula. You can also use the first one to get the answer.