Question

How do you find the exact values of \[\tan {{15}^{\circ }}\] using half angle formulas?

Answer 1

Hint: In this problem, we have to find the exact value of \[\tan {{15}^{\circ }}\] using the half angle formula. We know that basically half angle formulas are derived from the sum of sine, cosine and tangent identities. By using the half angle formula \[\tan \dfrac{\theta }{2}=\dfrac{1-\cos \theta }{\sin \theta }\], we can find the given tangent value. In this formula we have to find the theta value, and substitute the exact sine and cosine degree values, to get the exact value.

Complete step by step answer:
We know that the given angle is,\[\tan {{15}^{\circ }}\]
We also know that one of the half angle formulas is
\[\tan \dfrac{\theta }{2}=\dfrac{1-\cos \theta }{\sin \theta }\]
We know that the given degree is 15, where
\[\begin{align}
  & \Rightarrow \dfrac{\theta }{2}={{15}^{\circ }} \\
 & \Rightarrow \theta ={{30}^{\circ }} \\
\end{align}\]
Now we can apply the half angle formula for \[\tan {{15}^{\circ }}\] as we have \[\theta ={{30}^{\circ }}\] .
\[\Rightarrow \tan {{15}^{\circ }}=\dfrac{1-\cos {{30}^{\circ }}}{\sin {{30}^{\circ }}}\] …… (1)
We know that the value of \[\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}\] and \[\sin {{30}^{\circ }}=\dfrac{1}{2}\]
Applying the above values in expression (1), we get
\[\begin{align}
  & \Rightarrow \tan {{15}^{\circ }}=\dfrac{1-\dfrac{\sqrt{3}}{2}}{\dfrac{1}{2}} \\
 & \Rightarrow \tan {{15}^{\circ }}=\dfrac{\dfrac{2-\sqrt{3}}{2}}{\dfrac{1}{2}} \\
 & \Rightarrow \tan {{15}^{\circ }}=2-\sqrt{3} \\
\end{align}\]
 We know that the value of \[\sqrt{3}=1.732\], applying this value in the above step to get the exact value, we get
\[\Rightarrow \tan {{15}^{\circ }}=2-1.732=0.267\]

Therefore, the exact value of \[\tan {{15}^{\circ }}\] using half angle formula is 0.267.

Note: Students make mistakes while applying the sine and cosine values. We should know that to solve these types of problems, we have to know basic trigonometric identities, formulas, degree values and properties. We should also know that we have different methods to find the value using the half angle formula. We should also know some root values to be applied in these types of problems.