Question

How do you find the value of \[\cot \dfrac{\theta }{2}\] given \[\tan \theta = \dfrac{{ - 7}}{{24}}\] and \[\dfrac{{3\pi }}{2} < \theta < 2\pi \] ?

Answer 1

Hint: We are given one of the trigonometric functions that is \[\tan \theta = \dfrac{{ - 7}}{{24}}\]. Trigonometric functions are the ratios of sides of a right angled triangle. Tan function is the ratio of opposite side to adjacent side. Using the Pythagoras theorem we will find the remaining side that is hypotenuse of the triangle. Then we will use trigonometric identity to get the value of \[\cot \dfrac{\theta }{2}\].

Complete step by step solution:
First we will find the remaining side of the triangle so that we can find the remaining ratios of the trigonometric function that will help us to find the value of \[\cot \dfrac{\theta }{2}\].
We know that \[\text{hypotenuse} = \sqrt {\text{base}^2+ \text{height}^2} \]
Here base and height are the opposite and adjacent sides. So putting the values,
\[\text{hypotenuse} = \sqrt {{7^2} + {{24}^2}} \]
Taking the squares,
\[\text{hypotenuse} = \sqrt {49 + 576} \]

On adding we get,
\[\text{hypotenuse} = \sqrt {625} \]
Taking the root we get,
\[\text{hypotenuse} = 25\]
Now we have all the sides of the triangle.
We know that
\[\tan \dfrac{\theta }{2} = \dfrac{{1 - \cos \theta }}{{\sin \theta }}\]
So we need the sin and cos function.
\[\sin \theta = \dfrac{{\text{opp.side}}}{{\text{hypotenuse}}}\]
\[\Rightarrow\sin \theta = \dfrac{7}{{25}}\]

Since tangent function is given negative and angel falls in the fourth quadrant our sin and cosine function will also be signed according to the fourth quadrant.
So \[\sin \theta = \dfrac{{ - 7}}{{25}}\]
Then the next function
\[\cos \theta = \dfrac{{\text{adj.side}}}{{\text{hypotenuse}}}\]
\[\Rightarrow\cos \theta = \dfrac{{24}}{{25}}\]
Cos is positive in the fourth quadrant.
Now let’s get back to the identity
\[\tan \dfrac{\theta }{2} = \dfrac{{1 - \cos \theta }}{{\sin \theta }}\]
Putting the values,
\[\tan \dfrac{\theta }{2} = \dfrac{{1 - \dfrac{{24}}{{25}}}}{{\dfrac{{ - 7}}{{25}}}}\]

Taking LCM of numerator,
\[\tan \dfrac{\theta }{2} = \dfrac{{\dfrac{{25 - 24}}{{25}}}}{{\dfrac{{ - 7}}{{25}}}}\]
\[\Rightarrow\tan \dfrac{\theta }{2} = \dfrac{{\dfrac{1}{{25}}}}{{\dfrac{{ - 7}}{{25}}}}\]
Cancelling the same denominators,
\[\tan \dfrac{\theta }{2} = \dfrac{{ - 1}}{7}\]
Now we know that \[\cot \dfrac{\theta }{2} = \dfrac{1}{{\tan \dfrac{\theta }{2}}}\]
Putting the values,
\[\cot \dfrac{\theta }{2} = \dfrac{1}{{\dfrac{{ - 1}}{7}}}\]
On solving the ratio we get,
\[\therefore\cot \dfrac{\theta }{2} = - 7\]

Hence, the value of \[\cot \dfrac{\theta }{2}\] is - 7.

Note: We need not to find the angle of the function we only need to find the value. Also note that we are given the value of \[\tan \theta = \dfrac{{ - 7}}{{24}}\] so we cannot make the angle half or anything like that. So we need to use the multiple angle formula as mentioned above. So use the formula and don’t go to find the value of angle.