Problem 249 - Precalculus

Question

For the following exercises, use identities to simplify the expression. \(\frac{\sec t}{\csc t}\)

Step-by-Step Solution

Answer

The expression simplifies to \( \tan t \).

Step 1: Rewrite using identities

Start by rewriting the given expression, \( \frac{\sec t}{\csc t} \), using trigonometric identities. Recall that \( \sec t = \frac{1}{\cos t} \) and \( \csc t = \frac{1}{\sin t} \). Substitute these identities into the expression:\[\frac{\frac{1}{\cos t}}{\frac{1}{\sin t}}\]

Step 2: Simplify the fraction

When dividing by a fraction, multiply by the reciprocal. Thus, \( \frac{\frac{1}{\cos t}}{\frac{1}{\sin t}} \) becomes:\[\frac{1}{\cos t} \times \frac{\sin t}{1} = \frac{\sin t}{\cos t}\]

Step 3: Identify the simplified expression

Recognize that \( \frac{\sin t}{\cos t} \) is equivalent to the tangent function. Therefore,\[\frac{\sin t}{\cos t} = \tan t\]Thus, the original expression \( \frac{\sec t}{\csc t} \) simplifies to \( \tan t \).

Key Concepts

Secant

The secant function is a lesser-known trigonometric function related to the cosine function. In trigonometry, the secant of an angle, denoted as \( \sec \theta \), is the reciprocal of the cosine. That means if you know the value of the cosine of an angle, you can find the secant by taking the reciprocal, or in other words, dividing 1 by the cosine.
For example, mathematically, we express secant as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

The secant function is particularly useful in various math problems and in solving triangles, especially within the unit circle where the radius is 1. This identity allows you to rewrite expressions that have secant in terms of cosine, which can often be more familiar and easier to work with.

Cosecant

The cosecant function is another reciprocal trigonometric identity. It is related to the sine function. The cosecant of an angle, often represented by \( \csc \theta \), is the reciprocal of the sine of the angle. This means for any given angle, instead of dealing with sine, you can use its reciprocal, which is 1 divided by the sine. Mathematically, this can be written as:

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

Just like secant, cosecant is valuable in simplifying expressions or solving equations because it transforms expressions into a form that's more straightforward, using sine. Understanding cosecant can help you bridge more complex trigonometric problems to common ones using the sine function.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a key skill in trigonometry. It involves rewriting expressions in a simpler or more understandable form using known identities like secant and cosecant. As illustrated in the problem, by using the known identities to convert expressions, we make them more manageable.
Here's how you can simplify complex expressions:

• Identify and apply trigonometric identities. For example, replace \( \sec \theta \) with \( \frac{1}{\cos \theta} \) and \( \csc \theta \) with \( \frac{1}{\sin \theta} \).
• Simplify fractions by multiplying by the reciprocal, a fundamental property when dividing by fractions.
• Look for common identities in the resulting expressions, such as recognizing \( \frac{\sin \theta}{\cos \theta} \) as \( \tan \theta \).

Simplifying expressions can help not just in calculating values more easily, but also in understanding the relationships between different trigonometric functions more deeply. In our exercise, simplifying led from a ratio of secant to cosecant directly to the tangent function, showing how connected these functions can be.