Problem 46 - Precalculus with Calculus Previews

Question

Use the given information to find (a) \(\cos (x / 2)\), (b) \(\sin (x / 2)\), and (c) \(\tan (x / 2)\). $$ \cos x=\frac{4}{5}, 3 \pi / 2

Step-by-Step Solution

Answer

(a) \(\cos(x/2) = \frac{3\sqrt{10}}{10}\), (b) \(\sin(x/2) = \frac{\sqrt{10}}{10}\), (c) \(\tan(x/2) = \frac{1}{3}\).

Step 1: Find \\(x\\) in Quadrant IV

Given \(3\pi/2 < x < 2\pi\), we know that \(x\) is in the fourth quadrant where \(\cos x\) is positive and \(\sin x\) is negative. Given \(\cos x = \frac{4}{5}\), \(x\) is confirmed to be in Quadrant IV.

Step 2: Determine \\sin x\\ using \\cos x\\ and Pythagorean Identity

Using the identity \(\sin^2 x + \cos^2 x = 1\), substitute \(\cos x = \frac{4}{5}\):\[\sin^2 x = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25}.\]Therefore, \(\sin x = -\frac{3}{5}\) since \(x\) is in the fourth quadrant (where sine is negative).

Step 3: Use Half-Angle Formula for \\cos(x/2)\\

The formula for cosine of half an angle is \(\cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}}\). Substitute \(\cos x = \frac{4}{5}\):\[\cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \frac{4}{5}}{2}} = \pm \sqrt{\frac{9}{10}} = \pm \frac{3\sqrt{10}}{10}.\] Choose \(+\) as \(\frac{x}{2}\) is in the first quadrant.

Step 4: Use Half-Angle Formula for \\sin(x/2)\\

The formula for sine of half an angle is \(\sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{2}}\). Substitute \(\cos x = \frac{4}{5}\):\[\sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \frac{4}{5}}{2}} = \pm \sqrt{\frac{1}{10}} = \pm \frac{1}{\sqrt{10}} = \pm \frac{\sqrt{10}}{10}.\] Choose \(+\) as \(\frac{x}{2}\) is in the first quadrant.

Step 5: Use Half-Angle Formula for \\tan(x/2)\\

Compute \(\tan\left(\frac{x}{2}\right)\) using \(\sin\left(\frac{x}{2}\right)\) and \(\cos\left(\frac{x}{2}\right)\):\[\tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} = \frac{\frac{\sqrt{10}}{10}}{\frac{3\sqrt{10}}{10}} = \frac{1}{3}.\]

Key Concepts

Half-Angle Formulas

The half-angle formulas are powerful tools in trigonometry. These formulas help us calculate the cosine, sine, and tangent of half of a given angle, using easier-to-find values from the angle itself. These formulas are expressed as follows:

• For cosine: \( \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} \)
• For sine: \( \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{2}} \)
• For tangent: \( \tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} \)

These formulas require knowledge of the sign (positive or negative) for each quarter of the unit circle. This means we need to know which quadrant the final angle lies in to choose the right sign.
For half-angle problems like this, understanding the quadrant preferences makes all the difference, as they determine the expression's sign. A practical example of using these formulas is finding \( \cos(x/2) \), \( \sin(x/2) \), and \( \tan(x/2) \) from a known \( \cos x \) value.

Pythagorean Identity

The Pythagorean identity is one of the fundamental building blocks in trigonometry. It is defined as: \[ \sin^2 x + \cos^2 x = 1 \] This identity ensures that, for any angle \(x\), the square of the sine plus the square of the cosine equals one. It's analogous to the Pythagorean theorem from geometry.
In this particular exercise, where \( \cos x = \frac{4}{5} \), we used this identity to find \( \sin x \). First, substitute the known cosine value into the identity to get:
\[ \sin^2 x = 1 - \cos^2 x = 1 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} \] Since we're dealing with the fourth quadrant, where sine is negative, \( \sin x = -\frac{3}{5} \). This highlights how the Pythagorean identity is crucial for connecting the sine and cosine functions of an angle.

Trigonometric Functions in Quadrants

Understanding how trigonometric functions behave across the four quadrants of the unit circle is essential to solving many problems. Each quadrant has specific characteristics regarding the signs of its trigonometric functions.
Let's summarize the key points:

• Quadrant I: All trigonometric values are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

In this exercise, since \(x\) is in Quadrant IV (per the condition \(3\pi/2 < x < 2\pi\)), we know:

• \( \cos x = \frac{4}{5} \)
• \( \sin x = -\frac{3}{5} \)

When looking for half-angles, such as \(x/2\), we consider where these angles might land. Specifically, \(x/2\) often falls into Quadrant I in many standard trigonometric exercises, where all trigonometric functions are positive. This makes selecting the positive sign in half-angle formulas appropriate for our angle findings.