Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Write expression as a single trigonometric function or a power of a trigonometric function. (You may wish to use a graph to support your result.) $$\csc ^{2} t-1$$

Step-by-Step Solution

Answer

\( \cot^2 t \)

Step 1: Recall Trigonometric Identity

The given expression is \( \csc^2 t - 1 \). Recall the Pythagorean identity: \( \csc^2 t = 1 + \cot^2 t \). This identity will help us simplify the expression.

Step 2: Substitute the Identity into the Expression

Substitute \( \csc^2 t = 1 + \cot^2 t \) into the expression: \( \csc^2 t - 1 = (1 + \cot^2 t) - 1 \).

Step 3: Simplify the Expression

Simplify the expression: \( (1 + \cot^2 t) - 1 = \cot^2 t \). Therefore, the expression \( \csc^2 t - 1 \) simplifies to \( \cot^2 t \).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are essential in mathematics, especially when dealing with angles and triangles. These functions help us relate the angles of a triangle to its side lengths. The primary trigonometric functions are:

• Sine (\( \sin \theta \)) - Opposite side over the hypotenuse
• Cosine (\( \cos \theta \)) - Adjacent side over the hypotenuse
• Tangent (\( \tan \theta \)) - Opposite side over the adjacent side
• Cosecant (\( \csc \theta \)) - Reciprocal of sine, or hypotenuse over the opposite side
• Secant (\( \sec \theta \)) - Reciprocal of cosine, or hypotenuse over the adjacent side
• Cotangent (\( \cot \theta \)) - Reciprocal of tangent, or adjacent side over the opposite side

Each of these functions provides unique but related information about the relationship between the angles and sides of a triangle in trigonometry. For instance, the cosecant function is useful when you know the sine of an angle but need to find its relationship with the hypotenuse. Similarly, cotangent helps in understanding the ratio between the other two sides of the triangle. Remembering these functions and their reciprocals is vital for simplifying trigonometric expressions and solving problems.

The Pythagorean Identities in Trigonometry

The Pythagorean identities are fundamental equations in trigonometry that relate the squares of the main trigonometric functions. These identities derive from the Pythagorean theorem and help simplify expressions and solve trigonometric equations. The three main Pythagorean identities are:

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

These identities are incredibly useful for transforming and simplifying trigonometric expressions. They provide connections between different functions. For example, knowing one identity can help you convert between \( \tan \theta \) and \( \sec \theta \) or between \( \cot \theta \) and \( \csc \theta \).
Understanding and memorizing these identities allow you to see deeper relationships in trigonometric equations and can simplify otherwise complex calculations.

Trigonometric Simplification Techniques

Trigonometric simplification involves using identities and properties of trigonometric functions to make expressions easier to work with. This process is important in solving equations, evaluating functions, and making analyses more efficient. Here are some tips for simplifying trig expressions:

• Use basic identities like Pythagorean identities to change complex expressions into simpler ones.
• Look for opportunities to use reciprocal identities, such as turning \( \csc \theta \) into \( 1/\sin \theta \)
• Factor common factors when they appear in expressions.
• Cancel terms wherever applicable to reduce the complexity.

In the given exercise, we simplify the expression \( \csc^2 t - 1 \) using the identity \( \csc^2 t = 1 + \cot^2 t \). We substitute this identity into the expression, resulting in a simplified form, \( \cot^2 t \).
This process illustrates how turning an expression involving a difference into one using these identities can make problems easier to handle. Always remember, practice is key to mastering trigonometric simplification.