Question

If \[\sec \theta = 1\dfrac{1}{4}\] then calculate the value of \[\tan \dfrac{\theta }{2}\].
A. \[\dfrac{1}{3}\]
B. \[\dfrac{3}{4}\]
C. \[\dfrac{1}{4}\]
D. \[\dfrac{5}{4}\]

Answer 1

Hint: First we will calculate \[\cos \theta \] from the equation \[\sec \theta = 1\dfrac{1}{4}\]. Then we will apply the half-angle formula \[\tan \dfrac{\theta }{2} = \sqrt {\dfrac{{1 - \cos \theta }}{{1 + \cos \theta }}} \].

Formula used
\[\cos \theta = \dfrac{1}{{\sec \theta }}\]
Half angle formula:\[\tan \dfrac{\theta }{2} = \sqrt {\dfrac{{1 - \cos \theta }}{{1 + \cos \theta }}} \]

Complete step by step solution:
Given that \[\sec \theta = 1\dfrac{1}{4}\].
Rewrite the mixed fraction to improper fraction
\[\sec \theta = \dfrac{5}{4}\]
Now we will apply the formula \[\cos \theta = \dfrac{1}{{\sec \theta }}\] in above equation
\[\dfrac{1}{{\sec \theta }} = \dfrac{4}{5}\]
\[ \Rightarrow \cos \theta = \dfrac{4}{5}\]
Now we will apply half angle form to calculate \[\tan \dfrac{\theta }{2}\]
\[\tan \dfrac{\theta }{2} = \sqrt {\dfrac{{1 - \cos \theta }}{{1 + \cos \theta }}} \]
Putting \[\cos \theta = \dfrac{4}{5}\] in the above equation
\[\tan \dfrac{\theta }{2} = \sqrt {\dfrac{{1 - \dfrac{4}{5}}}{{1 + \dfrac{4}{5}}}} \]
\[ \Rightarrow \tan \dfrac{\theta }{2} = \sqrt {\dfrac{{\dfrac{1}{5}}}{{\dfrac{9}{5}}}} \]
Cancel out \[\dfrac{1}{5}\] from the numerator and denominator
\[ \Rightarrow \tan \dfrac{\theta }{2} = \sqrt {\dfrac{1}{9}} \]
\[ \Rightarrow \tan \dfrac{\theta }{2} = \dfrac{1}{3}\]
Hence option A is the correct option.

Additional information:
The half angle formulas are:
\[\sin \theta = 2\sin \dfrac{\theta }{2}\cos \dfrac{\theta }{2}\]
\[\cos \theta = 1 - 2{\sin ^2}\dfrac{\theta }{2}\]
\[\cos \theta = 2{\cos ^2}\dfrac{\theta }{2} - 1\]
\[\cos \theta = {\cos ^2}\dfrac{\theta }{2} - {\sin ^2}\dfrac{\theta }{2}\]

Note: Students often confused to apply the formulas \[\tan \dfrac{\theta }{2} = \sqrt {\dfrac{{1 - \cos \theta }}{{1 + \cos \theta }}} \] and \[\tan \dfrac{\theta }{2} = \sqrt {\dfrac{{1 + \cos \theta }}{{1 - \cos \theta }}} \]. But the correct formula is \[\tan \dfrac{\theta }{2} = \sqrt {\dfrac{{1 - \cos \theta }}{{1 + \cos \theta }}} \].