Problem 22 - Algebra and Trigonometry

Question

Simplify the trigonometric expression. $$ \frac{1+\cot A}{\csc A} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \sin A + \cos A \).

Step 1: Rewrite in terms of sinus and cosine

To simplify the expression \( \frac{1 + \cot A}{\csc A} \), we first express cotangent and cosecant in terms of sine and cosine. \( \cot A = \frac{\cos A}{\sin A} \) and \( \csc A = \frac{1}{\sin A} \). So the expression becomes \[ \frac{1 + \frac{\cos A}{\sin A}}{\frac{1}{\sin A}}. \]

Step 2: Simplify the numerator

Combine the terms in the numerator: \( 1 + \frac{\cos A}{\sin A} = \frac{\sin A}{\sin A} + \frac{\cos A}{\sin A} = \frac{\sin A + \cos A}{\sin A} \).

Step 3: Divide by the denominator using properties of fractions

The expression is \( \frac{\frac{\sin A + \cos A}{\sin A}}{\frac{1}{\sin A}} \). To divide by a fraction, multiply by its reciprocal. Thus: \[ \frac{\sin A + \cos A}{\sin A} \times \sin A = \sin A + \cos A. \]

Step 4: Final Simplification

Since the \( \sin A \) in the numerator and denominator cancels, we are left with \( \sin A + \cos A \). Thus, the simplified expression is \( \sin A + \cos A \).

Key Concepts

sine and cosine identities

Sine and cosine are fundamental trigonometric functions often used to redefine other trigonometric functions. The sine of an angle in a right triangle represents the ratio of the length of the side opposite the angle to the hypotenuse. The cosine represents the ratio of the length of the adjacent side to the hypotenuse. These two functions help form the basis for many identities in trigonometry.

A key identity involving sine and cosine is the Pythagorean identity:

• \( ext{sin}^2 A + ext{cos}^2 A = 1\)

This identity is useful because it allows relationships between different trigonometric functions to be established. By expressing the cotangent and cosecant in terms of sine and cosine in the provided exercise, we can seamlessly simplify the given expression.

trigonometric expressions

Trigonometric expressions are equations involving trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant. They are used extensively in mathematics, physics, and engineering to solve problems involving angles and distances.

Simplifying trigonometric expressions often involves rewriting them in terms of sine and cosine, as seen in the original exercise. This can help make the expression easier to work with or solve. Furthermore, recognizing patterns and utilizing common identities, such as the

• quotient identities (e.g., \( ext{tan} A = rac{ ext{sin} A}{ ext{cos} A}\))
• and reciprocal identities (e.g., \( ext{csc} A = rac{1}{ ext{sin} A}\)).

These can greatly aid in simplifying or solving for unknown values in a trigonometric expression.

cotangent and cosecant

Cotangent and cosecant are two less commonly used trigonometric functions, but they serve specific purposes in trigonometry. The cotangent (cot) of an angle is the reciprocal of the tangent, while the cosecant (csc) is the reciprocal of the sine.

They are defined as:

• \( ext{cot} A = rac{ ext{cos} A}{ ext{sin} A}\)
• \( ext{csc} A = rac{1}{ ext{sin} A}\)

In the original problem, expressing cotangent and cosecant in terms of sine and cosine was a crucial step in simplifying the expression. This approach often simplifies the problem because sine and cosine are more directly related through their identities. Mastery of these reciprocal relationships can provide a pathway to finding solutions more efficiently, especially in more complex trigonometric problems.