Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-22 :\) a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the value of the function value found in a. $$ \sin 86^{\circ} $$

Step-by-Step Solution

Answer

The function \( \\sin 86^\circ \\) in terms of its cofunction is \( \\cos 4^\circ \\, and its value is approximately \( 0.9976 \\).

Step 1: Understanding Cofunctions

Cofunctions are trigonometric functions that are related to each other. Specifically, the sine and cosine functions are cofunctions. The cofunction identity states that \( \sin\theta = \cos\left(90^\circ - \theta\right) \). We will apply this identity to our given function.

Step 2: Rewrite Using Cofunction Identity

We have the function \( \sin 86^\circ \. \) Using the cofunction identity \( \sin \theta = \cos (90^\circ - \theta) \, \) we rewrite \( \sin 86^\circ \) as \( \cos (90^\circ - 86^\circ) \). This simplifies to \( \cos 4^\circ \).

Step 3: Calculate the Value

Now we need to find the value of \( \cos 4^\circ \) to four decimal places. Using a calculator or trigonometric table, we find that \( \cos 4^\circ \approx 0.9976 \).

Key Concepts

Sine and Cosine Relationship

In the world of trigonometry, the sine and cosine functions share a special connection known as cofunction identities. These identities highlight the complementary nature of angles in trigonometric functions. Specifically, the cofunction identity for sine and cosine is expressed as \( \sin\theta = \cos(90^\circ - \theta) \). This reveals that the sine of an angle is equal to the cosine of its complement, which is the angle subtracted from \( 90^\circ \).

Understanding these relationships allows us to interchange sine and cosine effortlessly, making complex trigonometric calculations more manageable. Using the example, \( \sin 86^\circ \) becomes \( \cos 4^\circ \). Here, \( 90^\circ - 86^\circ \) gives us \( 4^\circ \), demonstrating the intermixed roles of sine and cosine when angles are involved. This insight can be a powerful tool for solving trigonometric problems and is pivotal in many applications of mathematics, such as engineering and physics.

Trigonometric Functions

Trigonometric functions are fundamental in understanding the relationships between the angles and sides of triangles. The primary trigonometric functions are sine, cosine, and tangent, each portraying a unique ratio of a right triangle’s sides. Sine, for instance, is defined as the ratio of the opposite side to the hypotenuse, while cosine is the ratio of the adjacent side to the hypotenuse.

These functions are essential not only in geometry but also in describing periodic phenomena such as sound waves, electromagnetism, and many other natural processes. When using functions like sine and cosine, cofunction identities become incredibly useful, especially in solving equations and transforming functions. By relating angles as complements, we can simplify expressions and find function values quickly, making calculations both swift and accurate.

Function Value Calculation

Calculating the value of a trigonometric function to a high degree of precision, such as four decimal places, is often necessary in scientific and engineering computations. In the example provided, we transform the sine value \( \sin 86^\circ \) using a cofunction into \( \cos 4^\circ \). With this transformation, the next step is to find \( \cos 4^\circ \) using a calculator or trigonometric table.

Navigating these calculations may involve:

• Finding the angle’s cosine value from known tables or software.
• Utilizing a scientific calculator set to degree mode to ensure accuracy.
• Applying the required precision by rounding off to four decimal places.

After performing these steps, we determine that \( \cos 4^\circ \) approximates to \( 0.9976 \). This precision is crucial in applications that demand high accuracy, including navigation systems, architecture, and aerospace engineering, where even minor errors in computation can lead to significant consequences.