Problem 24 - Precalculus Mathematics for Calculus

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: Express in terms of Sine and Cosine

Start by rewriting the trigonometric functions in terms of sine and cosine. Recall that \( \cot A = \frac{\cos A}{\sin A} \) and \( \csc A = \frac{1}{\sin A} \). Substitute these into the expression: \[\frac{1+\cot A}{\csc A} = \frac{1+\frac{\cos A}{\sin A}}{\frac{1}{\sin A}}\]

Step 2: Simplify the Expression

To simplify this expression, multiply the numerator by the reciprocal of the denominator. This becomes:\[\left( 1 + \frac{\cos A}{\sin A} \right) \times \sin A = ( \sin A + \cos A )\] The \( \sin A \) from the reciprocal of the denominator cancels out the division in the numerator.

Key Concepts

Simplifying Expressions

Simplifying trigonometric expressions might seem complex at first, but it gets easier with practice. The main idea is to manipulate the expression to reach its simplest form, where no further reduction is possible. The expression we examined was \( \frac{1+\cot A}{\csc A} \). The process of simplification involves rewriting it using fundamental trigonometric identities and basic arithmetic.

• First, identify the identities involved. Here, reciprocal identities can be used.
• Replace the trigonometric functions with their known equivalent expressions in terms of sine and cosine.
• Perform operations like addition, multiplication, and division, aiming to cancel terms when possible.

Breaking down the components and using identities consistently simplifies the work significantly. Remember to double-check each step to ensure accuracy in simplification.

Sine and Cosine

The sine and cosine functions are the backbone of trigonometry, forming the foundation for many identities and expressions. In the exercise, we used these functions by re-expressing \( \cot A \) and \( \csc A \) in terms of sine and cosine.

• The cotangent function \( \cot A \) is represented as \( \frac{\cos A}{\sin A} \). This relationship shows the ratio of the adjacent side to the opposite side in a right triangle.
• The cosecant function \( \csc A \) is equivalent to \( \frac{1}{\sin A} \). It represents the inverse of the sine function, indicating the reciprocal of the opposite side over the hypotenuse.

Understanding these relationships simplifies problems by reducing complex expressions into more familiar terms, allowing for standard arithmetic operations.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions play a crucial role when simplifying expressions. Each primary trigonometric function (sine, cosine, and tangent) has a corresponding reciprocal function: cosecant, secant, and cotangent respectively.

• \( \csc A = \frac{1}{\sin A} \)
• \( \sec A = \frac{1}{\cos A} \)
• \( \cot A = \frac{1}{\tan A} \)

Using these reciprocal functions can often simplify complex expressions more quickly and reveal hidden relationships among the trigonometric ratios. In our specific example, understanding \( \csc A \) as \( \frac{1}{\sin A} \) positioned the expression in a form that facilitated its simplification effortlessly. Remember, mastery over these functions can enhance problem-solving skills markedly by promoting a deeper comprehension of the connections between various trigonometric forms.