Question

What is \[\tan (\dfrac{\theta }{2})\] in terms of trigonometric functions of a unit $\theta $?

Answer 1

Hint: Trigonometry is a branch of mathematics that studies the relationship between the angles and the sides of a right-angled triangle. The relationship between sides and angles is established for six trigonometric functions.

Complete step by step solution:
A branch of mathematics that studies the relationship between the angles and the sides of a right-angled triangle. There are six trigonometric functions. Every trigonometric function can be represented in terms of the other trigonometric functions.
In trigonometry, the tangent function is a periodic function that is very useful. The trigonometric ratio between the adjacent side and the opposite side of a right triangle comprising that angle is called the tangent of an angle.
The identity $\tan \theta = \dfrac{{2\tan (\dfrac{\theta }{2})}}{{1 - {{\tan }^2}(\dfrac{\theta }{2})}}$,
Let us take $\tan (\dfrac{\theta }{2}) = x$, then the above equation will become:
$\tan \theta = \dfrac{{2x}}{{1 - {x^2}}}$
$ \Rightarrow tan\theta \times (1 - {x^2}) = 2x$
$ \Rightarrow - \tan \theta {x^2} - 2x + \tan \theta = 0$
$ \Rightarrow \tan \theta {x^2} + 2x - \tan \theta = 0$
Using the Quadratic formula, we get:
$x = \dfrac{{ - 2 \pm \sqrt {{2^2} - 4 \times \tan \theta \times ( - \tan \theta )} }}{{2\tan \theta }}$
$ \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {4 + 4 \times {{\tan }^2}\theta } }}{{2\tan \theta }}$
$ \Rightarrow x = \dfrac{{ - 2 \pm 2\sqrt {{{\sec }^2}\theta } }}{{2\tan \theta }}$
$ \Rightarrow x = \dfrac{{ - 2 \pm 2\sec \theta }}{{2\tan \theta }}$
$ \Rightarrow x = \dfrac{{ - 1 \pm \sec \theta }}{{\tan \theta }}$
Substituting $\tan (\dfrac{\theta }{2}) = x$ into the above equation, we get:
$ \Rightarrow \tan (\dfrac{\theta }{2}) = \dfrac{{ - 1 \pm \sec \theta }}{{\tan \theta }}$
Thus, \[\tan (\dfrac{\theta }{2})\] in terms of trigonometric functions of a unit $\theta $ is $\dfrac{{ - 1 \pm \sec \theta }}{{\tan \theta }}$.

Additional Information:
There are many uses for trigonometric functions in our real lives. It is used in oceanography to figure out how high the waves are in the seas. The sine and cosine functions are important in the study of periodic functions, which include sound and light waves. Trigonometry and Algebra make up Calculus.

Note:
The trigonometric functions sine and cosine are used to create each of the trigonometric functions in some way. The tangent of $x$ is said to be the sine of $x$ divided by the cosine of $x$.