Problem 20 - Algebra 2

Question

Given \(\cos \theta=-\frac{4}{5}\) and \(90^{\circ}<\theta<180^{\circ},\) find the exact value of each expression. $$ \cos \frac{\theta}{2} $$

Step-by-Step Solution

Answer

\(\cos(\frac{\theta}{2}) = \sqrt[]{0.1}\) or approximately \(0.316227\)

Step 1: Application of Half-Angle Formula

Apply the half-angle formula for cosine function: \(\cos(\frac{\theta}{2}) = \pm \sqrt[]{\frac{1+\cos(\theta)}{2}}\). Plugging the given values into the formula, you get \(\cos(\frac{\theta}{2}) = \pm \sqrt[]{\frac{1-4/5}{2}}\)

Step 2: Simplification

Simplify the equation to get \(\cos(\frac{\theta}{2}) = \pm \sqrt[]{\frac{1/10}} = \pm \sqrt[]{0.1}\)

Step 3: Determining the Sign

Since \(90^{\circ} < \theta < 180^{\circ}\), it means \(\frac{\theta}{2}\) will lie in the first quadrant (because \(45^{\circ} < \frac{\theta}{2} < 90^{\circ}\)). In the first quadrant the cosine function is positive. Therefore, \(\cos(\frac{\theta}{2}) = \sqrt[]{0.1}\)

Key Concepts

Half-Angle Formula

The Half-Angle Formula is a useful tool in trigonometry for finding the value of trigonometric functions of half angles. Specifically, when you are given a trigonometric function like cosine or sine for an angle, the half-angle formula can help in calculating the function of half of that angle.

For the cosine, the half-angle formula is expressed as:

• \( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \)

The formula involves a square root on the right side, indicating that the result will have a positive or negative sign. The sign is determined based on the quadrant in which the resulting angle \( \frac{\theta}{2} \) lies. This formula is particularly important in calculus and higher mathematics, where simplifying expressions is often necessary.

Cosine Function

The cosine function, denoted as \( \cos(\theta) \), is one of the fundamental trigonometric functions. It relates the adjacent side of a right triangle to its hypotenuse in a right-angled triangle and can also be viewed through the lens of the unit circle.

On the unit circle:

• \( \cos(\theta) \) corresponds to the x-coordinate of the point reached by the angle \( \theta \)

This means as the angle \( \theta \) changes, the value of \( \cos(\theta) \) moves between -1 and 1. In our original exercise, we know \( \cos(\theta) = -\frac{4}{5} \), representing a position in the second quadrant where cosine values are negative.

Such understanding is critical when dealing with trigonometric identities and solving equations involving angles.

First Quadrant

The first quadrant of the unit circle (0° to 90°) is a critical region for trigonometric evaluations. In this quadrant, all trigonometric functions have positive values.

Key points about the first quadrant include:

• \( \sin(\theta) > 0 \)
• \( \cos(\theta) > 0 \)
• \( \tan(\theta) > 0 \)

When \( \frac{\theta}{2} \) lies within the first quadrant, as in our problem, it's given that the angle \( \frac{\theta}{2} \) ranges from 45° to less than 90°, ensuring that \( \cos\left(\frac{\theta}{2}\right) \) must be positive. This information helps determine the correct sign when using the half-angle formula.

Angle Simplification

Angle simplification is a crucial step in trigonometric problem-solving. It's about breaking down complex angles into simpler or more manageable forms.

For example, in the given exercise, the original angle \( \theta \) is located between 90° and 180°, which can be cumbersome. By simplifying it to \( \frac{\theta}{2} \), the angle now falls into the first quadrant, where calculations can be cleaner and straightforward.

This process of simplification involves reducing larger angles into smaller measures (or half-angles) and can significantly ease the calculation process, especially in solving trigonometric equations or when integrating functions in calculus. By using half-angle identities, trigonometric expressions become simpler to evaluate.