Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Write each expression in terms of its co-function. $$sec $39^{\circ}$$

Step-by-Step Solution

Answer

\( \sec(39^\circ) = \csc(51^\circ) \)

Step 1: Understand Co-function Identity

In trigonometry, co-function identities relate the values of complementary angles. For secant, the co-function is cosecant. According to the co-function identity for secant, we have: \( \sec(\theta) = \csc(90^\circ - \theta) \).

Step 2: Calculate the Complementary Angle

Determine the complementary angle of \(39^\circ\). Remember, complementary angles sum up to \(90^\circ\). Therefore, calculate \(90^\circ - 39^\circ = 51^\circ\).

Step 3: Write the Co-function Expression

Using the complementary angle calculated in Step 2, apply the co-function identity. Thus, \( \sec(39^\circ) = \csc(51^\circ) \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a fundamental trigonometric function. It is the reciprocal of the cosine function. In simpler terms, secant of an angle \( \theta \) is equal to \( \frac{1}{\cos \theta} \).
\(\sec\) is very useful in trigonometry for solving various problems, including those involving right triangles and wave functions.

• The secant function helps measure the ratio of the hypotenuse to the adjacent side in a right-angle triangle.
• In mathematical calculations, it doubles as an essential tool in trigonometric identities and equations.

Remembering the relationships between various trig functions, such as secant and cosine, is crucial for simplifying complex expressions.

Cosecant Function

The cosecant function, represented as \( \csc \theta \), is another vital trigonometric function. It's defined as the reciprocal of the sine function, which means \( \csc \theta = \frac{1}{\sin \theta} \). While perhaps less commonly used than sine or cosine, it plays a key role in certain calculations, particularly those involving non-right angles.

• In right triangles, the cosecant relates the hypotenuse to the opposite side.
• Its usage extends to solving complex equations and identities by breaking down functions into simpler terms.

Recognizing how the cosecant function interacts with complementary angles is crucial for converting expressions from one form into another.

Complementary Angles

Complementary angles are pairs of angles whose measures sum up to \(90^{\circ}\). They are particularly significant in the study of trigonometry, where they link certain trigonometric functions called co-functions.

• By nature, if one angle in the pair is \( \theta \), the other must be \(90^{\circ} - \theta\).
• This relationship is crucial in identifying co-function identities such as \( \sec \theta = \csc(90^{\circ} - \theta) \) and vice versa.

Understanding complementary angles allows for the conversion of trigonometric functions into their co-functions, simplifying equations and aiding in the resolution of complex mathematical problems.