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Chapter 6: Problem 83

Show that \(\tan (x / 2)\) has the same sign as \(\sin x\) for any real number \(x\)

Short Answer

Expert verified
Tan(x/2) depends on the sin(x) sign with positive denominator 1 + cos(x)

Step by step solution

01

- Understand the problem

We need to show that \(\tan(x / 2)\) has the same sign as \(\tan(x / 2)\) has the same sign as \(\tan(x / 2)\) for any real number \(x\). This means we should prove that both are either positive or negative simultaneously.
02

- Expression for \(\tan(x / 2)\)

Recall the double-angle identity for tangent: \[\tan(x/2) = \frac{\sin x}{1 + \cos x}\] or \[\tan(x/2) = \frac{1 - \cos x}{\sin x}\]. These identities help relate \(\tan(x / 2)\) to \(\tan(x / 2)\)
03

- Consider the signs

To compare the signs, let's consider the identity \[\tan(x/2) = \frac{\sin x}{1 + \cos x}\] and analyze the signs of the numerator \(\sin x\) and the denominator \(1 + \cos x\). Note that \(1 + \cos x \) is always positive as \(\cos x\) ranges between -1 to 1.
04

- Compare the signs

Since \(1 + \cos x\) is always positive, the sign of \(\tan(x / 2)\) depends entirely on the sign of \(\tan(x / 2)\). Therefore, \(\tan(x/2)\) and \(\tan(x/2)\) will indeed always have the same sign because \(\sin x\) and \(\sin x\) are directly related and their quotient by a positive number preserves the sign.
05

- Conclusion

We have shown that \(\tan(x / 2)\) has the same sign as \(\sin(x / 2)\) for any real number \(x\). Both numerator of the fraction \(\sin(x/2) = \) and denominator \(1 + cos(x) = \) now have the same positive or negative sign

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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 \(\tan\), is one of the most fundamental trigonometric functions. It is defined as the ratio of the sine and cosine functions: \(\tan x = \frac{\sin x}{\cos x}\). This relationship plays a key role in many trigonometric identities and is particularly useful in analyzing angles and triangles.
sine function
The sine function, denoted as \(\sin\), is another essential trigonometric function. It represents the y-coordinate of a point on the unit circle as an angle is swept from the positive x-axis. For any angle \(x\), \(\sin x\) ranges between -1 and 1, inclusively. This function is used extensively in geometry, physics, and engineering.
double-angle identities
Double-angle identities help simplify trigonometric expressions involving angles that are double another angle. For instance, the double-angle identity for sine is \(\sin 2x = 2 \sin x \cos x\), and the double-angle identity for tangent is \(\tan 2x = \frac{2 \tan x}{1- \tan^2 x}\). These identities are essential in proving and working with trigonometric functions. They also simplify calculations and are used in solving trigonometric equations.

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Most popular questions from this chapter

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\) $$\cos (2 \alpha)=3 / 5 \text { and } 0^{\circ}<2 \alpha<90^{\circ}$$

Find all values of \(\alpha\) in \(\left[0^{\circ}, 360^{\circ}\right)\) that satisfy each equation. $$2 \sin \alpha=-\sqrt{3}$$

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\) $$\sin (2 \alpha)=-8 / 17 \text { and } 180^{\circ}<2 \alpha<270^{\circ}$$

Verify that each equation is an identity. $$\frac{\sin 4 t}{4}=\cos ^{3} t \sin t-\sin ^{3} t \cos t$$

Use identities to simplify each expression. Do not use a calculator. $$\frac{\sin 12^{\circ}}{1+\cos 12^{\circ}}$$
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