Problem 23 - A Graphical Approach to Precalculus with Limits

Question

A student writes "1 \(+\cot ^{2}=\csc ^{2}\) " Comment on this student's work.

Step-by-Step Solution

Answer

The student's expression is a correct trigonometric identity: \(1 + \cot^2(\theta) = \csc^2(\theta)\).

Step 1: Remembering Basic Trigonometric Identities

The student seems to refer directly to a basic trigonometric identity. Recall that one of the Pythagorean identities is: \[1 + \cot^{2}(\theta) = \csc^{2}(\theta)\] This identity is true for all angles where the functions are defined.

Step 2: Verifying the Identity

Start by expressing \( \cot(\theta) \) and \( \csc(\theta) \) in terms of sine and cosine:\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]\[ \csc(\theta) = \frac{1}{\sin(\theta)} \]Substitute these into the identity to confirm:\[1 + \left(\frac{\cos(\theta)}{\sin(\theta)}\right)^2 = \left(\frac{1}{\sin(\theta)}\right)^2\]

Step 3: Simplifying Each Side

Simplify both sides of the expression: For the left-hand side:\[1 + \frac{\cos^2(\theta)}{\sin^2(\theta)} = \frac{\sin^2(\theta) + \cos^2(\theta)}{\sin^2(\theta)}\]Since \(\sin^2(\theta) + \cos^2(\theta) = 1\), this becomes:\[ \frac{1}{\sin^2(\theta)} \] Matching the right-hand side, \( \csc^2(\theta) \), ensures they are equal.

Step 4: Conclusion

Both sides have simplified to \( \frac{1}{\sin^2(\theta)} \), meaning that the identity upholds. The student correctly referenced a true trigonometric identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental truths in trigonometry. They hold for all values of variables where the trigonometric functions are defined. These identities help explain relationships between different trigonometric functions. Among these, Pythagorean identities are widely used.

Pythagorean identities relate the square of the sine, cosine, tangent, cotangent, secant, and cosecant functions. A well-known example is:

• \( \ sin^2(\theta) + cos^2(\theta) = 1 \)
• \( \ 1 + tan^2(\theta) = sec^2(\theta) \)
• \( \ 1 + cot^2(\theta) = csc^2(\theta) \)

These identities are crucial for simplifying trigonometric expressions and solving problems. They provide the foundational techniques needed to transform complex trigonometric problems into simpler forms. Understanding them is essential for advancing further in mathematics.

Cotangent and Cosecant

Cotangent and cosecant are two primary trigonometric functions that often appear in various equations and identities. They are complementary to tangent and sine, respectively.

The **cotangent** function is the reciprocal of the tangent:

• \( cot(\theta) = \frac{cos(\theta)}{sin(\theta)} \)

It describes the ratio of the adjacent side to the opposite side in a right triangle. The cotangent function is particularly useful in problems involving angle measurements and periodic patterns.

The **cosecant** function is the reciprocal of the sine:

• \( csc(\theta) = \frac{1}{sin(\theta)} \)

Cosecant represents the hypotenuse to opposite side ratio in a right triangle. It often appears in trigonometric identities because of its reciprocal relationship with the sine function. Understanding these functions is critical in exploring the deeper connections in trigonometry.

Simplification of Expressions

Simplifying expressions is a key skill in mathematics and trigonometry. It involves reducing expressions to their simplest form while retaining their original value. This process often uses identities to transform and minimize expressions.

When given an expression like \( 1 + \cot^2(\theta) \), recognizing it as a Pythagorean identity allows us to simplify it directly to \( csc^2(\theta) \). This reduction uses basic identities such as \( sin^2(\theta) + cos^2(\theta) = 1 \) to show:

• \( \frac{sin^2(\theta) + cos^2(\theta)}{sin^2(\theta)} = \frac{1}{sin^2(\theta)} \)

This matches the cosecant squared, demonstrating both sides are equal. By expressing cotangent and cosecant in terms of sine and cosine, you can simplify more complex expressions. This process of simplification is vital in solving equations and understanding geometric relationships.