Problem 11 - Algebra and Trigonometry

Question

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \frac{\sec \theta-\cos \theta}{\sin \theta} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \tan \theta \).

Step 1: Rewrite Secant in Terms of Cosine

The secant function \( \sec \theta \) is the reciprocal of the cosine function. Therefore, we can rewrite it as \( \sec \theta = \frac{1}{\cos \theta} \). This allows us to rewrite the expression as: \[ \frac{\frac{1}{\cos \theta} - \cos \theta}{\sin \theta} \]

Step 2: Combine Terms in Numerator

To combine the terms in the numerator, find a common denominator which is \( \cos \theta \). Rewrite \( \cos \theta \) as \( \frac{\cos^2 \theta}{\cos \theta} \) so that the numerator becomes: \[ \frac{1}{\cos \theta} - \cos \theta = \frac{1 - \cos^2 \theta}{\cos \theta} \]

Step 3: Simplify Using Trigonometric Identity

We know the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), which can be rewritten as \( 1 - \cos^2 \theta = \sin^2 \theta \). Use this identity to simplify the expression in the numerator: \[ \frac{1 - \cos^2 \theta}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta} \]

Step 4: Simplify the Total Expression

Now place the simplified numerator over the original denominator \( \sin \theta \):\[ \frac{\sin^2 \theta}{\cos \theta} \times \frac{1}{\sin \theta} = \frac{\sin \theta}{\cos \theta} \]The expression \( \frac{\sin \theta}{\cos \theta} \) is equivalent to \( \tan \theta \).

Step 5: Final Simplified Expression

The trigonometric expression is simplified to \( \tan \theta \). This is the expression written solely in terms of \( \sin \theta \) and \( \cos \theta \) originally, but reduced to its simplest form.

Key Concepts

Sine and Cosine Relationship

In trigonometry, sine and cosine are two fundamental functions that describe the relationship between the angles and sides of a right triangle. They are often used as building blocks for more complex trigonometric expressions. Understanding their inter-connections is key to simplifying expressions efficiently.

Here are some essential points about the sine and cosine relationship:

• The sine of an angle, denoted as \( \sin \theta \), represents the ratio of the length of the side opposite the angle to the hypotenuse of the triangle.
• The cosine of an angle, written as \( \cos \theta \), describes the ratio of the adjacent side's length to the hypotenuse.
• Sine and cosine are complementary in that \( \sin(90^\circ - \theta) = \cos \theta \) and \( \cos(90^\circ - \theta) = \sin \theta \).

Understanding these relationships helps in rewriting expressions like \( \sec \theta = \frac{1}{\cos \theta} \) in terms of sine and cosine, which simplifies manipulation in more complex scenarios.

Trigonometric Simplification

Trigonometric simplification involves transforming an expression into a simpler, more digestible format. This process is useful for solving equations, integrating, and evaluating limits, among other applications.

Typically, it involves the following steps:

• Rewrite Exponential Trigonometric Functions: Convert reciprocal and quotient functions into their corresponding sine and cosine forms. For example, turning \( \sec \theta \) into \( \frac{1}{\cos \theta} \) simplifies further algebraic manipulation.
• Combine Like Terms: Use common denominators to combine terms, as performed in the original step when \( \sec \theta - \cos \theta \) was rewritten with \( \cos \theta \) as the common denominator.
• Apply Known Identities: Employ identities to replace complex expressions, like utilizing \( 1 - \cos^2 \theta = \sin^2 \theta \) from the Pythagorean identity, to further reduce complexity.

This step-by-step approach to simplification aids students in managing intricate expressions by breaking them down into simpler, familiar components.

Pythagorean Identities

The Pythagorean identities are among the most essential identities in trigonometry. They express the fundamental relationship between the sine and cosine functions, providing a powerful tool for simplification.

The core Pythagorean identity is:

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

This identity can be rearranged to express other useful forms, such as:

• \( 1 - \sin^2 \theta = \cos^2 \theta \)
• \( 1 - \cos^2 \theta = \sin^2 \theta \)

Using these identities makes it possible to transform expressions for simpler calculations. For example, in the original problem, \( 1 - \cos^2 \theta = \sin^2 \theta \) was used to convert the numerator, allowing the expression to be simplified to \( \tan \theta \).

Learning to apply these identities is a crucial step in mastering trigonometry, providing a solid foundation for tackling more complex equations and integrals.