Given $\cos(x) = -\frac{5}{13}$ and $x$ is in the second quadrant, find the value of $\tan(\frac{x}{2})$.
Understand the Problem
We are given the cosine of an angle $x$ and the quadrant in which it lies. We need to find the value of $ an( rac{x}{2})$. We can use the half-angle formula for tangent to solve this problem. Since $x$ is in the second quadrant, $ rac{x}{2}$ is in the first quadrant, so $ an( rac{x}{2})$ will be positive.
Answer
$ an(\frac{x}{2}) = \frac{3}{2}$
Answer for screen readers
$ an(\frac{x}{2}) = \frac{3}{2}$
Steps to Solve
- Write down the half-angle formula for tangent
The half-angle formula for tangent is given by: $$ an(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}$$ Also, we can use $$ an(\frac{x}{2}) = \frac{\sin x}{1 + \cos x}$$ or $$ an(\frac{x}{2}) = \frac{1 - \cos x}{\sin x}$$
- Determine the sign of $ an(\frac{x}{2})$
Since $x$ is in the second quadrant, we have $90^\circ < x < 180^\circ$. Therefore, $45^\circ < \frac{x}{2} < 90^\circ$, which means $\frac{x}{2}$ is in the first quadrant. In the first quadrant, tangent is positive, so $ an(\frac{x}{2})$ is positive. Thus, we will take the positive square root in the half-angle formula.
- Plug in the given value of $\cos x$ into the half-angle formula
We are given that $\cos x = -\frac{5}{13}$. Using the formula $ an(\frac{x}{2}) = \sqrt{\frac{1 - \cos x}{1 + \cos x}}$, we have: $$ an(\frac{x}{2}) = \sqrt{\frac{1 - (-\frac{5}{13})}{1 + (-\frac{5}{13})}} = \sqrt{\frac{1 + \frac{5}{13}}{1 - \frac{5}{13}}} = \sqrt{\frac{\frac{13+5}{13}}{\frac{13-5}{13}}} = \sqrt{\frac{\frac{18}{13}}{\frac{8}{13}}} = \sqrt{\frac{18}{8}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$$
$ an(\frac{x}{2}) = \frac{3}{2}$
More Information
The angle $x$ lies in the second quadrant, which means its value is between $90^\circ$ and $180^\circ$. When we take half of this angle, we get an angle between $45^\circ$ and $90^\circ$, which lies in the first quadrant. Tangent is positive in the first quadrant.
Tips
A common mistake is forgetting to consider the quadrant in which $x/2$ lies. This determines the sign of $ an(x/2)$. Also, mistakes in simplifying the fraction inside the square root can occur, so be careful with the arithmetic.
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