Problem 5 - Algebra 2

Question

Simplify each expression. $$ \csc \theta \cos \theta \tan \theta $$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Recall Trigonometric Identities

To simplify the expression \( \csc \theta \cos \theta \tan \theta \), we begin by recalling the trigonometric identities. \( \csc \theta = \frac{1}{\sin \theta} \), \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), and \( \cos \theta \) is the cosine function.

Step 2: Substitute Identities

Substitute the trigonometric identities into the original expression: \( \csc \theta = \frac{1}{\sin \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). The expression now becomes \( \frac{1}{\sin \theta} \times \cos \theta \times \frac{\sin \theta}{\cos \theta} \).

Step 3: Simplify the Expression

Now simplify the expression by canceling out terms. Notice that the \( \sin \theta \) in the numerator of \( \tan \theta \) cancels with \( \sin \theta \) in the denominator of \( \csc \theta \), and the \( \cos \theta \) in the numerator cancels with \( \cos \theta \) in the denominator. The expression simplifies to \( 1 \).

Key Concepts

Cosecant Function

The cosecant function is one of the six primary trigonometric functions. It's often not as commonly discussed as sine or cosine, but it's quite important. The cosecant, noted as \( \csc \theta \), is the reciprocal of the sine function. This means that:

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

This relationship implies that wherever sine is zero, cosecant is undefined due to division by zero. Therefore, it's important to remember that the domain of \( \csc \theta \) excludes those angles where \( \sin \theta = 0 \).
Some key points about the cosecant function include:

• Periodicity: Cosecant shares the same period as its sine counterpart, which is \( 2\pi \).
• Symmetry: Being an odd function, \( \csc(-\theta) = -\csc(\theta) \).

Understanding the cosecant function is crucial for simplifying trigonometric expressions, as seen in the given exercise.

Tangent Function

The tangent function, indicated as \( \tan \theta \), links the sine and cosine functions together. It's expressed as the ratio of sine to cosine:

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

This function provides valuable insight into the relationships between different trigonometric properties. The tangent function becomes undefined when \( \cos \theta = 0 \) because division by zero is not defined.
Key concepts about the tangent function include:

• Periodicity: The tangent function has a period of \( \pi \), meaning the pattern of the function repeats every \( \pi \) radians.
• Asymptotes: Because tangent is undefined wherever cosine equals zero, vertical asymptotes appear at odd multiples of \( \frac{\pi}{2} \).
• Symmetry: Like sine, tangent is an odd function, so \( \tan(-\theta) = -\tan(\theta) \).

These characteristics of the tangent function are useful, especially when simplifying compound trigonometric expressions.

Cosine Function

The cosine function is a fundamental part of trigonometry, known by the symbol \( \cos \theta \). It represents the x-coordinate of a point on the unit circle corresponding to an angle \( \theta \). The cosine function has several key features:

• Range: Its values are constrained between -1 and 1: \(-1 \leq \cos \theta \leq 1\).
• Periodicity: Like sine, its periodic cycle is \( 2\pi \), making it a repeating wave pattern.
• Symmetry: Being even, \( \cos(-\theta) = \cos(\theta) \), meaning the function is symmetrical about the y-axis.

The cosine function is integral in converting trigonometric expressions, especially those that combine several functions like in the original exercise. By understanding these key aspects of cosine, students can maneuver through complex problems with greater ease.