Problem 57 - Precalculus

Question

For the following exercises, simplify the given expression. $$ \sec \left(\frac{\pi}{2}-\theta\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( \csc \theta \).

Step 1: Identify the Trigonometric Identity

Recognize that the expression \( \sec \left( \frac{\pi}{2} - \theta \right) \) can be simplified using the co-function identities for trigonometric functions. Specifically, the co-function identity for secant is: \[ \sec \left( \frac{\pi}{2} - \theta \right) = \csc \theta.\] This step involves recalling the relationships between the trigonometric functions and their complements.

Step 2: Simplify the Expression

Using the co-function identity found in Step 1, simplify the expression:\[ \sec \left( \frac{\pi}{2} - \theta \right) = \csc \theta. \] This completes the simplification by using the identity to convert the secant of a complementary angle into cosecant.

Key Concepts

Co-function Identities in Trigonometry

Co-function identities are a fascinating aspect of trigonometry that connects one trigonometric function with another through complementary angles. A complementary angle means that the sum of two angles equals 90 degrees (or \(\frac{\pi}{2}\) radians in radian measure). These identities help simplify expressions and solve problems in trigonometry with ease.

Understanding co-function identities involves remembering a few key relationships between trigonometric pairs:

• Sine and cosine: \(\sin(\frac{\pi}{2} - \theta) = \cos(\theta)\) and \(\cos(\frac{\pi}{2} - \theta) = \sin(\theta)\)
• Tangent and cotangent: \(\tan(\frac{\pi}{2} - \theta) = \cot(\theta)\) and \(\cot(\frac{\pi}{2} - \theta) = \tan(\theta)\)
• Secant and cosecant: \(\sec(\frac{\pi}{2} - \theta) = \csc(\theta)\) and \(\csc(\frac{\pi}{2} - \theta) = \sec(\theta)\)

These identities show the reflections of trigonometric functions when angles are complementary, offering a powerful tool for simplifying trigonometric expressions.

Understanding Secant Function

The secant function is one of the six trigonometric functions, closely related to the cosine function. It is written as \(\sec(\theta)\) and is defined as the reciprocal of the cosine function:

\[\sec(\theta) = \frac{1}{\cos(\theta)}.\]

This means that the secant function represents the ratio of the hypotenuse to the adjacent side in a right triangle, assuming the angle \(\theta\) is measured with respect to the adjacent side. The secant function has some unique properties:

• It is undefined wherever the cosine function is zero, leading to vertical asymptotes in its graph.
• The secant function mirrors the periodicity of the cosine function, repeating every \(2\pi\) radians.

Recognizing and using the secant function's relationships, like co-function identities, can simplify complex problems in trigonometry.

The Role of Cosecant in Trigonometry

Cosecant, abbreviated as \(\csc\), is another key player in the world of trigonometry. It is the reciprocal of the sine function:

\[\csc(\theta) = \frac{1}{\sin(\theta)}.\]

In a right triangle, it represents the ratio of the hypotenuse to the opposite side for a given angle \(\theta\). Like other trigonometric functions, the cosecant has its own set of characteristics:

• It is undefined when the sine function equals zero, as division by zero is not possible.
• Similar to sine, the cosecant function is also periodic with a period of \(2\pi\) radians.

The cosecant function is particularly useful in solving problems involving co-function identities, as demonstrated in the example exercise with the simplification of \(\sec(\frac{\pi}{2} - \theta)\) which results in \(\csc(\theta)\). Understanding its role can greatly enhance solving trigonometric problems.