91Ó°ÊÓ

The tangent function, often written as \(\tan(x)\), is one of the fundamental functions in trigonometry. It is defined as the ratio of the sine function to the cosine function: \(\tan(x) = \frac{\text{{sin}}(x)}{\text{{cos}}(x)}\). In the context of the problem, we need to consider the tangent of half an angle, which has its own specific identity. For this exercise, the relevant identity is: \[ \tan \frac{x}{2} = \frac{\text{sin}(x)}{1 + \text{cos}(x)}\]. This identity helps us understand the relationship between \(\tan(x/2)\) and the sine function, simplifying our analysis of their signs.

The sine function, denoted as \(\text{sin}(x)\), is another core trigonometric function. It represents the y-coordinate of a point on the unit circle as the angle \x\ varies. The sine function is periodic with a period of \2\frac{\text{\tm}}\text\rad and ranges between \(-1, 1)\). In our problem, we are analyzing the sign of \(\text{sin}(x)\), which changes depending on the quadrant in which the angle lies:

• In the 1st and 2nd quadrants, \(\text{sin}(x)\) is positive
• In the 3rd and 4th quadrants, \(\text{sin}(x)\) is negative

Trigonometric proofs involve showing that certain trigonometric identities or properties hold true using established trigonometric functions and their properties. For our exercise, the goal was to prove that \(\tan(x/2)\) has the same sign as \(\text{sin}(x)\) for any angle \x\. Using the trigonometric identity \(\tan \frac{x}{2} = \frac{\text{sin}(x)}{1 + \text{cos}(x)}\) and noting that the denominator \(\text{1 + cos}(x)\) is always positive, we can conclude that the sign of \(\tan(x/2)\) must be the same as the sign of the numerator \(\text{sin}(x)\). This completes the proof and ensures that students understand how to relate different trigonometric functions using their identities.