Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Write each expression in terms of its co-function. $$\sin 38^{\circ} 29^{\prime}$$

Step-by-Step Solution

Answer

\( \sin 38^{\circ} 29^{\prime} = \cos 51^{\circ} 31^{\prime} \).

Step 1: Understanding the Co-function Identity

Trigonometric co-functions are complementary. This means that for any angle \( \theta \), the sine and cosine functions are related through the identity \( \sin(90^\circ - \theta) = \cos(\theta) \). Our task is to express the given sine function as a cosine function by using this identity.

Step 2: Identifying the Complementary Angle

To apply the co-function identity, we need to calculate the complementary angle for \( 38^{\circ} 29^{\prime}\). This can be done by subtracting the given angle from \( 90^\circ \). We perform the subtraction: \[ 90^{\circ} - 38^{\circ} 29^{\prime} = 51^{\circ} 31^{\prime}. \]

Step 3: Writing the Expression in Terms of the Co-function

Using our result from the previous step, we can write \( \sin 38^{\circ} 29^{\prime} \) in terms of its co-function. According to the co-function identity, we have \( \sin(38^{\circ} 29^{\prime}) = \cos(51^{\circ} 31^{\prime}) \).

Key Concepts

Understanding Co-functions

Co-functions are pairs of trigonometric functions that are related through complementary angles. The term "co-function" arises because these function pairs complete each other to form 90 degrees; hence, they are complementary. A key identity to remember is that the sine of an angle is equal to the cosine of its complementary angle. Specifically, this identity is expressed as:

• \( \sin(90^\circ - \theta) = \cos(\theta) \)
• \( \cos(90^\circ - \theta) = \sin(\theta) \)

This identity is incredibly useful because it allows us to convert any trigonometric expression from sine to cosine, or vice versa, by simply examining their complement relationship. In other words, these identities tell us how to transform between a function and its co-function using complementary angles.

The Role of Complementary Angles

Complementary angles are two angles whose measures add up to exactly 90 degrees. When working with trigonometric expressions, complementary angles help us transform and simplify these expressions. If you know the measure of one angle, you can easily find its complementary angle by subtracting from 90 degrees.Understanding this transformation is crucial in trigonometry because it demonstrates how certain angular relationships inherently tie functions together. This allows you to exploit the relationships between these angles when solving trigonometric problems. For example:

• Given \( \theta = 38^\circ 29' \), its complementary angle is \( 90^\circ - 38^\circ 29' = 51^\circ 31' \).

This calculation ensures that you can apply co-function identities effectively. Understanding complementary angles simplifies computations and aids in memorizing trigonometric identities.

Expressing Trigonometric Functions

Expressing trigonometric functions using their co-functions is a common technique in mathematics. This is often done to simplify a problem or to change function types between sine and cosine when necessary. Knowing how to transform expressions using co-function identities gives you a solid foundation for analyzing and solving trigonometric equations.To express a trigonometric function as its co-function:

• First, identify if your given angle has a complementary angle by subtracting from 90 degrees.
• Then, use the co-function identity. For example, for sine, transform using \( \sin(\theta) = \cos(90^\circ - \theta) \).
• As in our problem: \( \sin(38^\circ 29') = \cos(51^\circ 31') \)

This technique helps in transforming expressions, solving complex trigonometric identities, and is a vital skill for higher-level math studies, including calculus and geometry.