Problem 72 - Precalculus with Limits

Question

In Exercises 67-72, (a) determine the quadrant in which \( u/2 \) lies, and (b) find the exact values of \( \sin(u/2) \), \( \cos(u/2) \), and \( \tan(u/2) \) using the half-angle formulas. \( \sec u = \dfrac{7}{2}, \dfrac{3\pi}{2} < u < 2\pi \)

Step-by-Step Solution

Answer

The exact values of the half-angles are: \( \sin(u/2) = \sqrt{\frac{5}{14}} \), \( \cos(u/2) = -\sqrt{\frac{9}{14}} \), and \( \tan(u/2) = \sqrt{\frac{5}{9}} \). And \( u/2 \) lies in the 2nd quadrant.

Step 1: Determine the quadrant of \( u/2 \)

Given the relation \( \frac{3\pi}{2} < u < 2\pi \), the division of each part by 2 gives \( \frac{3\pi}{4} < \frac{u}{2} < \pi \). This shows that \( u/2 \) is located in the 2nd quadrant.

Step 2: Calculate \( \cos u \)

We know \( \sec u = \frac{7}{2} \), and since \( \sec u = \frac{1}{\cos u} \), this allows us to find \( \cos u = \frac{2}{7} \).

Step 3: Apply half-angle formulas for each trigonometric function

For half-angle formulas, the sine, cosine and tangent of \( u/2 \) are computed as follows: \( \sin(u/2) = \pm \sqrt{\frac{1 - \cos u}{2}} = \sqrt{\frac{1 - 2/7}{2}} = \sqrt{\frac{5}{14}} \), \( \cos(u/2) = \pm \sqrt{\frac{1 + \cos u}{2}} = \sqrt{\frac{1 + 2/7}{2}} = \sqrt{\frac{9}{14}} \), and \( \tan(u/2) = \frac{\sin(u/2)}{\cos(u/2)} = \frac{\sqrt{\frac{5}{14}}}{\sqrt{\frac{9}{14}}} = \sqrt{\frac{5}{9}} \). In the 2nd quadrant, \( \sin \) and \( \tan \) are positive, and \( \cos \) is negative, so we choose the positive root for \( \sin \) and \( \tan \), and the negative root for \( \cos \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics that help simplify expressions, solve equations, and prove relationships involving trigonometric functions. One of the key sets of identities involves half-angle identities, which are particularly useful when finding the sine, cosine, or tangent of half an angle.

To find the trigonometric functions for half angles, we use specific formulas:

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

These identities allow us to break down more complex angles into simpler calculations, especially when working with quadrants to determine the signs of the results.

Secant Function

The secant function, denoted as \( \sec \), is the reciprocal of the cosine function. In trigonometry, it is defined as \( \sec u = \frac{1}{\cos u} \).

This relationship hints at how knowing either \( \sec u \) or \( \cos u \) can help us find the other. For example, given \( \sec u = \frac{7}{2} \), we can deduce that \( \cos u = \frac{2}{7} \).

The secant function is particularly useful in various applications, including solving problems involving right triangles and simplifying expressions. Since secant is related to the cosine, it's important to remember that when \( \cos u \) approaches zero, \( \sec u \) tends toward infinity, highlighting their reciprocal nature.

Quadrants of Angles

Understanding the quadrants of an angle is crucial in determining the sign of trigonometric functions. The coordinate plane is divided into four quadrants: 1st, 2nd, 3rd, and 4th, and they are counterclockwise from the positive x-axis.

Each quadrant affects the sign of trigonometric functions:

• 1st Quadrant: All trigonometric functions are positive.
• 2nd Quadrant: Sine is positive; cosine and tangent are negative.
• 3rd Quadrant: Tangent is positive; sine and cosine are negative.
• 4th Quadrant: Cosine is positive; sine and tangent are negative.

When the problem states \( \frac{3\pi}{4} < \frac{u}{2} < \pi \), it places \( \frac{u}{2} \) in the 2nd quadrant. Knowing this helps in selecting the correct sign for half-angle formula values: positive for sine and tangent, negative for cosine.

Exact Trigonometric Values

Exact trigonometric values are values that can be represented without approximation. They often involve square roots or fractions and are preferred in mathematics for precision.

Using half-angle formulas enables us to express trigonometric values exactly. For instance, with \( \cos u = \frac{2}{7} \), one can find:

• \( \sin(\frac{u}{2}) = \sqrt{\frac{5}{14}} \)
• \( \cos(\frac{u}{2}) = -\sqrt{\frac{9}{14}} \)
• \( \tan(\frac{u}{2}) = \sqrt{\frac{5}{9}} \)

These exact values give us insights without rounding errors, ensuring high precision in mathematical problems. In the 2nd quadrant, the sign conventions state that only cosine takes the negative value, hence influencing the selection of signs for calculating trigonometric functions.