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Magazine Chapter 6: Problem 125
Determine whether each statement is true or false. $$\text { If } \tan x>0, \text { then } \tan \left(\frac{x}{2}\right)>0$$
Short Answer
Expert verified
The statement is False.
Step by step solution
01
Understanding the Assertion
The problem asks whether if \( \tan x > 0 \), it implies \( \tan\left(\frac{x}{2}\right) > 0 \) as well. We must analyze the behavior of the tangent function and its half-angle identity.
02
Analyze the Tangent Function
Recall that \( \tan x \) is positive in the first and third quadrants, i.e., when \( x \) is in the intervals \( (0, \pi) \) and \( (\pi, 2\pi) \) respectively.
03
Apply Half-Angle Formula
The tangent half-angle identity is: \[ \tan\left(\frac{x}{2}\right) = \frac{1 - \cos x}{\sin x} \]Using this identity, the sign of \( \tan\left(\frac{x}{2}\right) \) will depend on the signs of \( \sin x \) and \( \cos x \).
04
Quadrant Analysis for Half-Angle
Considering \( x \) in the interval \( (0, \pi) \), both \( \sin x > 0 \) and \( \cos x > 0 \). Thus, \( \frac{1 - \cos x}{\sin x} > 0 \).However, for \( x \) in \( (\pi, 2\pi) \), \( \tan x > 0 \) but \( \sin x < 0 \) and \( \cos x < 0 \), thus \( \frac{1 - \cos x}{\sin x} < 0 \).
05
Conclusion
From Step 4, it's evident that \( \tan\left(\frac{x}{2}\right) > 0 \) only if \( x \) is in \( (0, \pi) \). But if \( x \) is in \( (\pi, 2\pi) \), then \( \tan\left(\frac{x}{2}\right) < 0 \). Hence the statement is false.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Tangent Function
The tangent function, denoted as \( \tan x \), is a fundamental component of trigonometry. It's defined as the ratio of the sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \). Understanding the properties of the tangent function is crucial to analyzing different angles.
- The tangent function has a period of \( \pi \), which means it repeats every \( \pi \) radians.
- \( \tan x \) is positive when both \( \sin x \) and \( \cos x \) share the same sign, occurring in the first and third quadrants.
- In the first quadrant, \( \sin x > 0 \) and \( \cos x > 0 \), leading to a positive \( \tan x \).
- In the third quadrant, \( \sin x < 0 \) and \( \cos x < 0 \), which also results in a positive tangent.
- Interestingly, the tangent function can take any real value, ranging from negative to positive infinity.
Half-Angle Identities
Half-angle identities are vital in trigonometry as they allow us to find the tangent of half an angle using known angles. The tangent's half-angle identity is written as:\[\tan\left(\frac{x}{2}\right) = \frac{1 - \cos x}{\sin x}\]This identity helps in simplifying problems where angles are halved.
- The sign of \( \tan\left(\frac{x}{2}\right) \) is determined by the signs of \( \sin x \) and \( \cos x \).
- Given \( \sin x \) and \( \cos x \), you can substitute them into the formula to find the half-angle tangent.
Quadrant Analysis
Analyzing which quadrant an angle resides in is crucial for determining the sign of trigonometric functions. The tangent function's positivity depends on this quadrant location. Here's a breakdown:
- The first quadrant is where both sine and cosine are positive, leading to a positive tangent.
- In the second quadrant, sine is positive, but cosine is negative, giving a negative tangent.
- The third quadrant sees both sine and cosine negative, hence a positive tangent.
- Lastly, the fourth quadrant has a positive cosine and negative sine, leading to a negative tangent.
