Problem 14 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \sin \theta \).

Step 1: Rewrite tan using its definition

Recall that the trigonometric identity \( \tan \theta \) is equivalent to \( \sin \theta / \cos \theta \). We will substitute this into the expression. So \( \cos \theta \tan \theta \) becomes \( \cos \theta * (\sin \theta / \cos \theta)\)

Step 2: Simplify the expression

Now, the \( \cos \theta \) in the numerator and denominator cancel each other out, leaving us with just \( \sin \theta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that describe relationships between the trigonometric functions. These identities are crucial in simplifying and solving various trigonometric expressions and equations. In our example, we use the identity for the tangent function.

• The tangent function, \( \tan \theta \), is defined as \( \frac{\sin \theta}{\cos \theta} \).
• This basic identity stems from the fundamental relationships within a right triangle, where sine and cosine are the ratios of the sides.

Understanding trigonometric identities allows us to convert complex expressions into simpler forms, making it easier to handle mathematical problems.

Trigonometric Simplification

Trigonometric simplification involves reducing trigonometric expressions to their simplest form. It often includes the use of identities to eliminate terms, factor expressions, or cancel elements. In our example, we see this simplification process unfold with the expression \( \cos \theta \tan \theta \).To simplify, we:

• Utilize the identity for \( \tan \theta \) as \( \frac{\sin \theta}{\cos \theta} \).
• Substitute this identity into the original expression, transforming it into \( \cos \theta ( \frac{\sin \theta}{\cos \theta} ) \).
• Cancel the \( \cos \theta \) in the numerator and denominator, leading us to the simplified result of \( \sin \theta \).

This demonstrates how simplification can remove complexity and reveal a clearer solution.

Cosine Function

The cosine function is one of the primary trigonometric functions, essential in both pure and applied mathematics. It is often associated with the adjacent side and hypotenuse in a right triangle. Key characteristics include:

• It describes the ratio of the adjacent side to the hypotenuse: \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).
• It has a range between -1 and 1.
• Cosine is used extensively in formulas and identities, such as Pythagorean and co-function identities.

In our example, the cosine function initially appears in the expression \( \cos \theta \tan \theta \), serving as the means to simplify the equation when paired with the tangent identity.

Sine Function

The sine function is another fundamental trigonometric function, pivotal in calculations related to triangles and oscillating functions. As a central part of the trigonometric family, its defining characteristics include:

• Sine relates to the opposite side and hypotenuse: \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \).
• Its value also ranges from -1 to 1.
• The sine function helps in modeling waves and circular motion, illustrated in applications like sound and light waves.

In the context of our example, \( \sin \theta \) emerges as the simplified form of the expression \( \cos \theta \tan \theta \). This simplification highlights sine's role in unifying and simplifying trigonometric problems.