Problem 12 - Contemporary Precalculus

Question

Factor the given expression. $$\tan t \cos t+\cos ^{2} t$$

Step-by-Step Solution

Answer

Answer: The factored form of the given expression is $$\sin t + \cos^2 t$$.

Step 1: Simplify the expression

We can rewrite the tangent function as the ratio of sine to cosine: $$\frac{\sin t}{\cos t}\cos t + \cos^2 t$$

Step 2: Multiply and simplify the trigonometric expressions

Now, multiply the sine and cosine expressions in the first term: $$\sin t + \cos^2 t$$

Step 3: Factor out the common term

In this case, there is no common term to factor out from the expression. The simplified expression is: $$\sin t + \cos^2 t$$

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial for simplifying expressions and solving trigonometry problems. These identities are equations that hold true for all values of the variables involved. They serve as the foundational tools for transforming and rewriting trigonometric expressions into equivalent forms.

One of the most basic identities is the Pythagorean identity, which is expressed as \( \sin^2 t + \cos^2 t = 1 \). This particular identity connects the sine and cosine functions, and it is often used to express one function in terms of the other.

Another set of identities relates to the tangent function. For instance, one can express tangent as the ratio of sine and cosine: \( \tan t = \frac{\sin t}{\cos t} \). These relationships are essential when encountering expressions involving multiple trigonometric functions as they allow us to simplify complex terms into a more manageable form.

Simplifying Expressions

The process of simplifying trigonometric expressions often involves a combination of algebraic manipulation and the strategic use of trigonometric identities. When simplifying, the goal is to transform the expression into its simplest form, which typically means having the fewest terms and the simplest possible functions.

To simplify an expression like \( \tan t \cos t + \cos^2 t \), you can start by breaking down complex functions into more basic ones, such as rewriting tangent in terms of sine and cosine. Then, look for common factors, and combine like terms where possible. You might also need to perform operations such as multiplication or division to consolidate the expression.

For example,

• Rewrite complex functions in terms of basic ones
• Factor out common terms when possible
• Combine like terms to reduce the expression
• Use specific identities to eliminate functions and simplify further

It's all about making the expression more approachable and easier to work with.

Tangent and Cosine Relationship

The relationship between tangent and cosine is a key aspect of simplifying trigonometric expressions involving these functions. As mentioned earlier, tangent can be expressed as the ratio of sine to cosine, \( \tan t = \frac{\sin t}{\cos t} \). This relationship is particularly helpful when faced with expressions that contain both tangent and cosine terms.

Knowing this relationship allows us to transform a product of tangent and cosine into a simpler expression involving sine. For instance, \( \tan t \cos t \) can be simplified by substitifying the ratio form for tangent, resulting in \( \frac{\sin t}{\cos t} \cos t \), which further simplifies to \( \sin t \). Now, the expression involves only sine and cosine separately rather than their product.

The transformation of \( \tan t \cos t + \cos^2 t \) to \( \sin t + \cos^2 t \) highlights the benefit of understanding these relationships—making otherwise complex expressions easier to interpret and solve.