Solving Trigonometry Problems: Finding Values

This problem asks us to find the value of a trigonometric expression given the value of a related expression. We are given the value of \( \frac{{(1 - \cos \theta )}}{{\sin \theta }} \) and need to find the value of \( \frac{{(1 + \cos \theta )}}{{\sin \theta }} \).

Understanding the Given Information

We are given the equation:

\( \frac{{(1 - \cos \theta )}}{{\sin \theta }} =  \frac{1}{5} \)

We need to find the value of:

\( \frac{{(1 + \cos \theta )}}{{\sin \theta }} \)

Using Trigonometric Identities

Let's consider the two expressions:

• Expression 1: \( \frac{1 - \cos \theta}{\sin \theta} \)
• Expression 2: \( \frac{1 + \cos \theta}{\sin \theta} \)

Notice the similarity between the numerators \( (1 - \cos \theta) \) and \( (1 + \cos \theta) \). This structure often suggests using the difference of squares formula, \( (a-b)(a+b) = a^2 - b^2 \), and the fundamental trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \), which can be rearranged as \( 1 - \cos^2 \theta = \sin^2 \theta \).

Step-by-Step Solution

Let the value we need to find be \( x \). So, \( x = \frac{{(1 + \cos \theta )}}{{\sin \theta }} \). We are given that \( \frac{{(1 - \cos \theta )}}{{\sin \theta }} = \frac{1}{5} \).

Let's multiply the two expressions together:

\( \left( \frac{{(1 - \cos \theta )}}{{\sin \theta }} \right) \times \left( \frac{{(1 + \cos \theta )}}{{\sin \theta }} \right) \)

Using the formula for multiplying fractions, multiply the numerators and the denominators:

\( =  \frac{{(1 - \cos \theta )(1 + \cos \theta )}}{{\sin \theta \cdot \sin \theta }} \)

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

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

Now, use the identity \( 1 - \cos^2 \theta = \sin^2 \theta \):

\( =  \frac{{\sin^2 \theta}}{{\sin^2 \theta }} \)

Assuming \( \sin \theta \neq 0 \), the expression simplifies to:

\( =  1 \)

So, we have found that:

\( \left( \frac{{(1 - \cos \theta )}}{{\sin \theta }} \right) \times \left( \frac{{(1 + \cos \theta )}}{{\sin \theta }} \right) = 1 \)

We know that \( \frac{{(1 - \cos \theta )}}{{\sin \theta }} = \frac{1}{5} \). Substitute this value into the equation:

\( \frac{1}{5} \times \left( \frac{{(1 + \cos \theta )}}{{\sin \theta }} \right) = 1 \)

Let \( x = \frac{{(1 + \cos \theta )}}{{\sin \theta }} \). The equation becomes:

\( \frac{1}{5} \times x = 1 \)

To find \( x \), multiply both sides of the equation by 5:

\( x = 1 \times 5 \)

\( x = 5 \)

Therefore, the value of \( \frac{{(1 + \cos \theta )}}{{\sin \theta }} \) is 5.

Summary of Calculation

The calculated value for \( \frac{{(1 + \cos \theta )}}{{\sin \theta }} \) is 5.

Revision Table: Key Trigonometry Concepts

Additional Information: Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a fundamental skill in trigonometry. It often involves using identities to rewrite expressions in a simpler form. Some common techniques include:

• Converting everything to sine and cosine.
• Using Pythagorean identities like \( \sin^2 \theta + \cos^2 \theta = 1 \), \( 1 + \tan^2 \theta = \sec^2 \theta \), and \( 1 + \cot^2 \theta = \csc^2 \theta \).
• Using reciprocal identities (e.g., \( \sec \theta = 1/\cos \theta \), \( \csc \theta = 1/\sin \theta \), \( \cot \theta = 1/\tan \theta \)).
• Using quotient identities (e.g., \( \tan \theta = \sin \theta / \cos \theta \), \( \cot \theta = \cos \theta / \sin \theta \)).
• Factoring expressions (like difference of squares, perfect squares).
• Finding a common denominator when adding or subtracting fractions.

In this problem, multiplying the two expressions was the key step because it allowed us to use the difference of squares and the Pythagorean identity to simplify the product to a constant (1).