Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function value. $$\sqrt{\frac{1+\cos 165^{\circ}}{1-\cos 165^{\circ}}}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \cot 82.5^{\circ} \).

Step 1: Identify the relevant trigonometric identity

The expression is of the form \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} \). This can be rewritten using the identity \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} = \cot \left( \frac{\theta}{2} \right) \).

Step 2: Calculate \( \frac{165^{\circ}}{2} \)

Divide 165 degrees by 2 to find the angle for the cotangent identity: \( \frac{165^{\circ}}{2} = 82.5^{\circ} \).

Step 3: Apply the identity to simplify the expression

Substitute \( \theta = 165^{\circ} \) into the identity: \( \sqrt{\frac{1 + \cos 165^{\circ}}{1 - \cos 165^{\circ}}} = \cot 82.5^{\circ} \).

Key Concepts

Understanding Cosine

The cosine function is one of the fundamental trigonometric functions. It relates an angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. When you see the term "cosine", it is often denoted as \( \cos \theta \), where \( \theta \) represents an angle.

Important properties of cosine include:

• Cosine values range between -1 and 1.
• \( \cos 0^{\circ} = 1 \) and \( \cos 90^{\circ} = 0 \).
• It is an even function, meaning \( \cos(-\theta) = \cos \theta \).

The key role of cosine in trigonometric identities allows us to simplify complex expressions by leveraging its properties. For example, in the given exercise, we analyze an expression containing \( \cos 165^{\circ} \) and use identities to transform it into a more manageable form.
By understanding how cosine interacts within identities, we can solve trigonometric problems more efficiently.

Demystifying Cotangent

Cotangent, abbreviated as \( \cot \), is another important trigonometric function. It is the reciprocal of the tangent function. This means \( \cot \theta = \frac{1}{\tan \theta} \), and it can also be expressed as the ratio of cosine over sine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Cotangent's main characteristics include:

• It is undefined for angles where \( \sin \theta = 0 \), such as \( \theta = 0^{\circ} \), \( 180^{\circ} \) and so on.
• \( \cot 45^{\circ} = 1 \).
• It is periodic with a period of \( 180^{\circ} \).

In the context of the exercise, the expression was simplified using the identity \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} = \cot \left( \frac{\theta}{2} \right) \). Understanding how cotangent works allows us to convert complex trigonometric forms into simpler expressions, such as converting the original square root expression into \( \cot 82.5^{\circ} \).

Simplification Using Trigonometric Identities

Angle simplification in trigonometry often relies on identities, which are essential tools for rewriting trigonometric expressions. In many cases, simplifying an angle makes its trigonometric function easier to evaluate or visualize.

One of the crucial identities we used in the exercise is:

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

This identity is particularly useful when dealing with expressions involving the square root of fractions with cosine terms. Simplifying the angle by dividing it, in this case 165 degrees, by 2, allows us to handle the trigonometric function more straightforwardly. The result is the cotangent of a smaller angle, \( 82.5^{\circ} \), which is often easier to comprehend.
By understanding and applying these identities, you can reduce complex expressions to simpler functions, facilitating problem solving in trigonometry.