Problem 47 - Precalculus with Calculus Previews

Question

Use the given information to find (a) \(\cos (x / 2)\), (b) \(\sin (x / 2)\), and (c) \(\tan (x / 2)\). $$ \tan x=2, \pi

Step-by-Step Solution

Answer

\(\cos(x/2) = -\sqrt{\frac{1 - \sqrt{1/5}}{2}}, \sin(x/2) = \sqrt{\frac{1 + \sqrt{1/5}}{2}}, \tan(x/2) = \sqrt{\frac{1 + \sqrt{1/5}}{1 - \sqrt{1/5}}}\).

Step 1: Understand the Interval and Find Quadrant

The interval \(\pi < x < 3\pi/2\) indicates that angle \(x\) is in the third quadrant. In this quadrant, \(\tan x\) is positive, while \(\sin x\) and \(\cos x\) are both negative.

Step 2: Use Identity for \(\sin x\) and \(\cos x\) Using \(\tan x\)

Since \(\tan x = 2\), use the identity \(\tan x = \frac{\sin x}{\cos x}\). Assume \(\sin x = -a\) and \(\cos x = -b\) with \(\frac{a}{b} = 2\). Solving gives \(a = 2b\), and using \(\sin^2 x + \cos^2 x = 1\), substitute \(a\) and \(b\): \((2b)^2 + b^2 = 1\). This yields \(5b^2 = 1\) or \(b = \pm \sqrt{\frac{1}{5}}\). Since \(x\) is in the third quadrant, \(\cos x = -\sqrt{\frac{1}{5}}\) and \(\sin x = -\sqrt{\frac{4}{5}}\).

Step 3: Find \(\cos(x/2)\) Using Half-Angle Identity

The half-angle identity is \(\cos(x/2) = \pm \sqrt{\frac{1 + \cos x}{2}}\). Substitute \(\cos x = -\sqrt{\frac{1}{5}}\) to find \(1 + \cos x = 1 - \sqrt{\frac{1}{5}}\). Therefore, \(\cos(x/2) = \pm \sqrt{\frac{1 - \sqrt{\frac{1}{5}}}{2}}\). Since \(x\) is in the third quadrant, \(x/2\) is in the second quadrant where \(\cos(x/2) < 0\). Thus, \(\cos(x/2) = -\sqrt{\frac{1 - \sqrt{\frac{1}{5}}}{2}}\).

Step 4: Find \(\sin(x/2)\) Using Half-Angle Identity

The half-angle identity is \(\sin(x/2) = \pm \sqrt{\frac{1 - \cos x}{2}}\). Use \(\cos x = -\sqrt{\frac{1}{5}}\) to find \(1 - \cos x = 1 + \sqrt{\frac{1}{5}}\). Therefore, \(\sin(x/2) = \pm \sqrt{\frac{1 + \sqrt{\frac{1}{5}}}{2}}\). Since \(x/2\) is in the second quadrant, \(\sin(x/2) > 0\), meaning \(\sin(x/2) = \sqrt{\frac{1 + \sqrt{\frac{1}{5}}}{2}}\).

Step 5: Find \(\tan(x/2)\) Using Half-Angle Identity

The half-angle identity for tangent is \(\tan(x/2) = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\). Substitute \(\cos x = -\sqrt{\frac{1}{5}}\) to find: \(1 - \cos x = 1 + \sqrt{\frac{1}{5}}\) and \(1 + \cos x = 1 - \sqrt{\frac{1}{5}}\). Therefore, \(\tan(x/2) = \pm \sqrt{\frac{1 + \sqrt{\frac{1}{5}}}{1 - \sqrt{\frac{1}{5}}}}\). Since in the second quadrant \(\tan(x/2) > 0\), \(\tan(x/2) = \sqrt{\frac{1 + \sqrt{\frac{1}{5}}}{1 - \sqrt{\frac{1}{5}}}}\).

Key Concepts

Half-Angle Identities

Half-angle identities are powerful tools in trigonometry. They let us find the trigonometric values of an angle that's half the size of an original angle. This is especially helpful when the original angle is not one of the typical angles we memorize, like 0°, 30°, 45°, etc.

For half-angle identities, we use formulas for sine, cosine, and tangent. These are:

• \(cos(x/2) = \pm \sqrt{\frac{1 + \cos x}{2}}\)
• \(sin(x/2) = \pm \sqrt{\frac{1 - \cos x}{2}}\)
• \(tan(x/2) = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\)

The choice of the sign (+ or -) depends on the quadrant of the angle \(x/2\). Each function's identity starts with a square root which means we consider both positive and negative roots.

However, the quadrant tells us which sign to use, so it's essential to understand the angle's position.

Trigonometric Functions

Trigonometric functions describe relationships between the angles and sides of triangles, particularly in right-angled triangles. The three primary functions are sine (\(\sin \)), cosine (\(\cos \)), and tangent (\(\tan \)). These fundamental functions have reciprocal functions, too: cosecant (\(\csc \)), secant (\(\sec \)), and cotangent (\(\cot \)).

Each of these functions can express a ratio of two sides of a right triangle:

• \(sin(x) = \frac{\text{opposite side}}{\text{hypotenuse}}\)
• \(cos(x) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)
• \(tan(x) = \frac{\text{opposite side}}{\text{adjacent side}}\)

The trigonometric functions are cyclical and repeat their values in a predictable pattern. These properties are vital for analyzing periodic phenomena.

In this exercise, first, we find \(\sin x\) and \(\cos x\) using \(\tan x = 2\) and employ the identity \(\tan x = \frac{\sin x}{\cos x}\). This equation helps rewrite the system in solvable terms.

Third Quadrant

In the unit circle, angles are divided into four quadrants. The third quadrant covers angles where \(\pi < x < \frac{3\pi}{2}\). Here, both the sine and cosine functions are negative, whereas the tangent function is positive. This affects the calculations of half-angle identities significantly.

When determining values for \(x/2\), we need to know the related quadrant. For \(x\) in the third quadrant, \(x/2\) falls in the second quadrant. This is because the angle measure halves but still spans across a different range: \(\frac{\pi}{2} < x/2 < \pi\), making \(\sin(x/2)\) positive and \(\cos(x/2)\) negative.

Understanding quadrant locations allows us to select the correct sign for the trigonometric values. For this exercise, evaluating half-angle identities using these insights helps find the precise values of the trigonometric functions.