Problem 72 - Algebra and Trigonometry

Question

Rewrite each expression as a simplified expression containing one term. $$ \frac{\cos (\alpha-\beta)+\cos (\alpha+\beta)}{-\sin (\alpha-\beta)+\sin (\alpha+\beta)} $$

Step-by-Step Solution

Answer

The simplified expression containing one term is \(\cot(b)\).

Step 1: Break Down the Numerator

Applying the trigonometric identities to the numerator, we get: \(\cos(a - b) + \cos(a + b) = \cos(a)\cos(b) + \sin(a)\sin(b) + \cos(a)\cos(b) - \sin(a)\sin(b) = 2\cos(a)\cos(b)\).

Step 2: Break Down the Denominator

We also apply the identities to the denominator: \(-\sin(a - b) + \sin(a + b) = -(\sin(a)\cos(b) - \cos(a)\sin(b)) + (\sin(a)\cos(b) + \cos(a)\sin(b)) = 2\cos(a)\sin(b)\).

Step 3: Simplify the Fraction

By substituting the results from Steps 1 and 2 into the original expression, we get: \(\frac{2\cos(a)\cos(b)}{2\cos(a)\sin(b)}\). This simplifies to \(\frac{\cos(b)}{\sin(b)} = \cot(b)\).

Key Concepts

Simplifying Expressions

Simplifying expressions, especially in trigonometry, is about transforming complex expressions into simpler forms. This process often involves using mathematical identities or properties to reduce or condense multiple terms into a single term.

• Identify and factor common terms wherever possible. This helps in reducing the complexity of the expression.
• Utilize trigonometric identities like Pythagorean identities, angle sum/difference identities, or double angle formulas to replace parts of the expression with a simpler term.
• For fractions, always look for opportunities to cancel out terms in the numerator and denominator.

In the exercise, simplification involved using trigonometric identities for cosine and sine. By facturing out common terms, the given complex fraction was reduced to its simplest form, the cotangent of an angle. Understanding how to apply these steps allows you to tackle similar problems with ease.

Trigonometry

Trigonometry is a field of mathematics that studies the relationships between the angles and sides of triangles. Primarily, it deals with ratios derived from circles, which leads to a better understanding of various physical and abstract components in science and engineering.

• Trigonometric functions such as sine, cosine, and tangent relationships are crucial. These functions help in understanding angle properties and solving complicated geometric problems.
• Knowing key identities like angle sum, difference identities, and double angle identities allows you to break down and analyze expressions effectively.
• These identities are particularly useful in simplifying expressions. They provide a shorthand mechanism to transform complex trigonometric forms into simpler equivalents.

In the given exercise, these trigonometric identities were used to break down the expressions involving cosine and sine, converting them into a more manageable form. Proper knowledge of trigonometric principles thus plays an essential role in such transformations.

Cotangent

The cotangent is a lesser-known but equally important trigonometric function. It's the reciprocal of the tangent and is defined as the ratio of the adjacent side to the opposite side in a right triangle.

• The cotangent function is given by the formula: \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). This fraction form was crucial in the step-by-step simplification of the expression presented in the exercise.
• By understanding the cotangent's relationship with cosine and sine, it is easier to simplify expressions involving these functions.
• In applied mathematics, cotangent finds its place in solving real-world problems where angles need to be interpreted, like inclined planes or wave calculations.

The final simplified expression in the exercise highlights the utility of the cotangent, converting a more complicated fraction into this standard trigonometric form. Mastering cotangent and its associated identities paves the way for efficiently solving advanced trigonometric challenges.