Question

Cofunctions. Express as a function of the complementary angle. $$\csc 82.7^{\circ}$$

Short answer

\( \csc(82.7^\circ) = \sec(7.3^\circ) \)

Step-by-step solution

Understanding the Cofunction Identity for cosecant

The cofunction identity for cosecant states that \( \csc(x) = \sec(90^\circ - x) \) for an angle \( x \) measured in degrees. This is because the cosecant of an angle is the reciprocal of the sine of that angle, and the sine of an angle is equal to the cosine of its complement.

Applying the Cofunction Identity

To express \( \csc(82.7^\circ) \) as a function of its complementary angle, use the identity \( \csc(\theta) = \sec(90^\circ - \theta) \). The complementary angle to \( 82.7^\circ \) is \( 90^\circ - 82.7^\circ = 7.3^\circ \). Therefore, \( \csc(82.7^\circ) = \sec(7.3^\circ) \).

Final Expression

The final expression is \( \sec(7.3^\circ) \), which is \( \csc(82.7^\circ) \) expressed as a function of the complementary angle.

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental to understanding and solving various mathematical problems involving angles and their relationships.
These identities are often equations that relate different trigonometric functions to each other.

Understanding Cofunction Identities

Among these identities, cofunction identities are particularly useful. They express the relationship between the trigonometric functions of complementary angles. An angle is complemented by another if the sum of both is 90 degrees, or \(\pi/2\) radians. The cofunction identities include statements like \(\sin(\theta) = \cos(90^\circ - \theta)\), implying that the sine of any angle is equal to the cosine of its complement.

This set of relationships also reflects how the functions secant, cosecant, tangent, and cotangent are related to their counterparts. For example, understanding that \(\tan(\theta) = \cot(90^\circ - \theta)\) simplifies the process of finding the value of one trigonometric function by knowing the value of another.

Secant Function

In trigonometry, the secant function, denoted as \(\sec\), is one of the six principal trigonometric functions. It is defined as the reciprocal of the cosine function, which means \(\sec(\theta) = \frac{1}{\cos(\theta)}\).

Relevance in Cofunction Identities

Within the scope of cofunction identities, the secant has a special relationship with the cosecant function. According to the identities, \(\sec(\theta) = \csc(90^\circ - \theta)\). This equation allows us to transform a problem involving cosecant into one involving secant, which may be more familiar or convenient to work with in certain scenarios.
Understanding the secant function and its behavior is essential, especially when we want to express trigonometric expressions as a function of complementary angles, as it allows us to navigate between these interlinked functions more smoothly.

Complementary Angles

Complementary angles are a pair of angles where the sum of their measures equals 90 degrees.

Significance in Trigonometry

The concept of complementary angles is essential in trigonometry because many trigonometric properties and identities are built upon the complementary relationship between angles. When we say that two angles are complementary, it indicates that the trigonometric functions of one angle will have a direct relationship to the cofunctions of the other angle.
This principle is applied in problems involving cofunction identities, like in our exercise where understanding that \(82.7^\circ\) and \(7.3^\circ\) are complementary allows us to rewrite \(\csc(82.7^\circ)\) as \(\sec(7.3^\circ)\). Cofunction identities thus provide a powerful tool to simplify complex trigonometric expressions and calculations by harnessing the inherent symmetries in the unit circle.