Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\cos \frac{x}{2}, \text { given } \sin x=-\frac{4}{5} \text { and } \frac{3 \pi}{2} < x < 2 \pi$$

Step-by-Step Solution

Answer

\(\cos \frac{x}{2} = \sqrt{\frac{4}{5}}\)

Step 1: Identify the Identity

We need to use the half-angle identity for cosine, which is given by: \[ \cos \frac{x}{2} = \pm \sqrt{\frac{1+\cos x}{2}} \]

Step 2: Determine the Quadrant of \(x\)

The information given is that \(\frac{3\pi}{2} < x < 2\pi\), which means \(x\) is in the fourth quadrant. In the fourth quadrant, cosine is positive.

Step 3: Calculate \(\cos x\) Using \(\sin x\)

Since \(\sin x = -\frac{4}{5}\), we can use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). This gives us: \[ \left(-\frac{4}{5}\right)^2 + \cos^2 x = 1 \] solving this, we get \( \cos x = \frac{3}{5} \) as cosine is positive in the fourth quadrant.

Step 4: Substitute \(\cos x\) into the Half-angle Identity

Substitute \(\cos x = \frac{3}{5}\) into the half-angle identity: \[ \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \frac{3}{5}}{2}} = \pm \sqrt{\frac{8}{10}} = \pm \sqrt{\frac{4}{5}} \]

Step 5: Determine the Sign of \(\cos \frac{x}{2}\)

Since \(\cos \frac{x}{2}\) is in the first or fourth quadrant when \(\frac{3\pi}{2} < x < 2\pi\), it is positive. Therefore, \( \cos \frac{x}{2} = \sqrt{\frac{4}{5}} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are integral to understanding the relations of angles and sides in triangles. The primary trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each of these functions can be thought of as a ratio of sides in a right triangle:

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

Additionally, transformations like reciprocal, product, and quotient identities can be derived for these functions, expanding their utility in complex calculations. Understanding these functions is crucial for delving into more advanced concepts like half-angle identities, which allow us to calculate the value of trigonometric functions for half angles, a task that can simplify many trigonometric equations.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry. It arises from the Pythagorean theorem, which deals with right-angled triangles. For any angle \(x\), the Pythagorean identity is given by:
\[ \sin^2 x + \cos^2 x = 1 \]
This identity illustrates how sine and cosine relate to each other. It can be rearranged to find one function if you know the other:

• For \(\sin x\), \(\cos^2 x = 1 - \sin^2 x\)
• For \(\cos x\), \(\sin^2 x = 1 - \cos^2 x\)

The Pythagorean identity is extremely useful, as shown in the step-by-step solution. For instance, when given \(\sin x = -\frac{4}{5}\), we used this identity to find \(\cos x\), determining it as \(\frac{3}{5}\) since cosine is positive in the specified quadrant. This identity helps us solve for unknown values, bridging gaps in calculations.

Unit Circle

The unit circle is a powerful visual aid in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle in standard position, where the angle's vertex is at the origin. From any point on the unit circle:

• \(x\)-coordinate equals \(\cos \theta\)
• \(y\)-coordinate equals \(\sin \theta\)

The beauty of the unit circle comes from its ability to represent these functions continuously, making it easier to understand complex trigonometric ideas with simple geometric relationships. This continues into more advanced topics such as half-angle identities, where the unit circle can help visualize the division of an angle, providing a clearer understanding of its cosine or sine value at half its measure. It serves as a constant reminder of the cyclic nature of these functions and their symmetries.

Quadrants in Trigonometry

The trigonometric functions change their signs depending on the quadrant in which the angle lies. The coordinate plane is divided into four quadrants:

• 1st Quadrant: Both \(\sin x\) and \(\cos x\) are positive.
• 2nd Quadrant: \(\sin x\) is positive, \(\cos x\) is negative.
• 3rd Quadrant: Both \(\sin x\) and \(\cos x\) are negative.
• 4th Quadrant: \(\sin x\) is negative, \(\cos x\) is positive.

Knowing which quadrant an angle is in greatly aids in determining the signs of trigonometric functions. For instance, with \( \frac{3\pi}{2} < x < 2\pi \), angle \(x\) lies in the fourth quadrant. This knowledge was crucial when applying the half-angle identity to find \(\cos \frac{x}{2}\). Recognizing that cosine remains positive in the fourth quadrant helped us ascertain that \(\cos x = \frac{3}{5}\) and ultimately that \(\cos \frac{x}{2}\) was positive, yielding \(\sqrt{\frac{4}{5}}\) as the solution.