Problem 20 - Algebra 2 and Trigonometry

Question

In \(15-22,\) write each given expression in terms of sine and cosine and express the result in simplest form. \(\frac{\sec \theta}{\csc \theta}\)

Step-by-Step Solution

Answer

The simplest form of the expression is \( \tan \theta \).

Step 1: Rewrite Secant and Cosecant in terms of Sine and Cosine

The secant function is defined as the reciprocal of the cosine function, i.e., \( \sec \theta = \frac{1}{\cos \theta} \). Similarly, the cosecant function is the reciprocal of the sine function, i.e., \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Substitute the Definitions into the Expression

Substitute the expressions for secant and cosecant into the original expression: \( \frac{\sec \theta}{\csc \theta} = \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}} \).

Step 3: Simplify the Fraction

The expression is a complex fraction. Simplify it by multiplying the numerator by the reciprocal of the denominator: \( \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}} = \frac{1}{\cos \theta} \times \frac{\sin \theta}{1} = \frac{\sin \theta}{\cos \theta} \).

Step 4: Express the Result in Simplest Form

The expression \( \frac{\sin \theta}{\cos \theta} \) can be simplified to \( \tan \theta \) since the tangent function is defined as the ratio of sine to cosine.

Key Concepts

Secant Reciprocal

The secant function is an important trigonometric function closely associated with the cosine function. It is defined as the reciprocal of cosine. This means:

• If you know the value of cosine, you can easily find secant by taking the reciprocal.
• The formula is: \( \sec \theta = \frac{1}{\cos \theta} \).

This relationship is valuable because some trigonometric problems are easier to solve when expressed in terms of sine and cosine. Knowing these reciprocal identities allows for straightforward conversions and simplifications. As you practice trigonometry, it becomes evident how frequently these relationships appear.

Cosecant Reciprocal

Similar to secant, the cosecant function is another reciprocal identity in trigonometry. It is defined as the reciprocal of the sine function:

• The formula to remember is: \( \csc \theta = \frac{1}{\sin \theta} \).
• When dealing with expressions involving cosecant, converting it to a sine form simplifies the computations.

Just like with secant, understanding the concept of reciprocal identities opens the door to more easily tackling trigonometric expressions. Recognizing these patterns can ease one's way through complex problems, especially in simplifying or solving equations.

Simplifying Trigonometric Expressions

When it comes to working with trigonometric expressions, simplification is a key skill. Understanding how to rewrite expressions using basic trigonometric identities can drastically reduce complexity. Here’s a quick guide:

• Identify reciprocal identities like secant and cosecant and rewrite them in terms of sine and cosine.
• Look for opportunities to combine terms by finding common denominators or factoring.
• When dealing with fractions, remember to cancel common factors or multiply by the reciprocal when needed.

In the solution process presented, we took the complex fraction \( \frac{\sec \theta}{\csc \theta} \) and ultimately simplified it to \( \tan \theta \) by using these strategies. Simplification often leads to more recognizable forms which one can easily interpret or manipulate further.

Tangent Function

The tangent function is an essential trigonometric function that stems from the sine and cosine functions. It is defined as the ratio of sine to cosine:

• Mathematically, this is expressed as: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Because tangent is a ratio, any trigonometric expression involving sine and cosine might simplify to tangent.
• Recognizing when an expression simplifies to tangent can be key in problem-solving.

In our exercise example, the expression \( \frac{\sin \theta}{\cos \theta} \) directly simplifies to \( \tan \theta \). Realizing this helps to drastically simplify calculations and see the relationship between different trigonometric identities.