Problem 8 - 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. $$ \sec 75^{\circ} $$

Step-by-Step Solution

Answer

\( \sec 75^{\circ} = \csc 15^{\circ} \approx 3.8637 \)

Step 1: Understanding Cofunction Identities

Cofunction identities relate trigonometric functions whose angles are complementary. For secant, the cofunction identity is given as \( \sec(\theta) = \csc(90^{\circ} - \theta) \). This means that for any angle \( \theta \), the secant of that angle equals the cosecant of its complement.

Step 2: Applying the Cofunction Identity

We are given \( \sec 75^{\circ} \). Using the cofunction identity: \[ \sec 75^{\circ} = \csc(90^{\circ} - 75^{\circ}) \] Simplifying inside the parentheses gives \( \csc(15^{\circ}) \). Therefore, \( \sec 75^{\circ} \) can be rewritten as \( \csc 15^{\circ} \).

Step 3: Calculating the Cosecant Value

To find the value of \( \csc 15^{\circ} \) to four decimal places, recognize that \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, calculate \( \sin(15^{\circ}) \) and use it to find \( \csc(15^{\circ}) \): \[ \csc(15^{\circ}) = \frac{1}{ \sin(15^{\circ}) } \].

Step 4: Finding the Sine of 15 Degrees

The exact value of \( \sin(15^{\circ}) \) can be calculated using the angle subtraction identity \( \sin(15^{\circ}) = \sin(45^{\circ} - 30^{\circ}) \). Applying the identity, \[ \sin(15^{\circ}) = \sin(45^{\circ}) \cos(30^{\circ}) - \cos(45^{\circ}) \sin(30^{\circ}) \]. \( \sin(45^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \), \( \sin(30^{\circ}) = \frac{1}{2} \), and \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \). Substitute these values to get: \[ \sin(15^{\circ}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4} \].

Step 5: Calculating Cosecant Using Sine Value

Given \( \sin(15^{\circ}) = \frac{\sqrt{6} - \sqrt{2}}{4} \), find \( \csc(15^{\circ}) \) by taking the reciprocal: \[ \csc(15^{\circ}) = \frac{1}{\sin(15^{\circ})} = \frac{4}{\sqrt{6} - \sqrt{2}} \]. To approximate \( \csc(15^{\circ}) \), use a calculator to find the decimal value. \( \csc(15^{\circ}) \approx 3.8637 \) to four decimal places.

Key Concepts

Secant and Cosecant Relationship

Understanding the relationship between the secant and cosecant functions requires familiarity with cofunction identities. Cofunction identities link trigonometric functions involving complementary angles, meaning two angles whose sum is 90 degrees. For example, the identity for the secant function is expressed as:

• \( \sec(\theta) = \csc(90^{\circ} - \theta) \)

This formula signifies that the secant of any angle \( \theta \) is equal to the cosecant of its complementary angle \( 90^{\circ} - \theta \). This relationship is particularly useful when rewriting trigonometric functions in terms of their cofunctions, as it allows the conversion from one function to another, simplifying calculations or deriving important trigonometric values when dealing with complex angles.

Trigonometric Complementary Angles

Complementary angles are two angles whose measurements add up to 90 degrees. They are a fundamental concept in trigonometry because they allow the interchanging of certain trigonometric functions through cofunction identities. Let's look at a few basic pairs:

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

Using these identities makes it easier to solve trigonometric equations by converting one trigonometric function to another, depending on the value of the angle. In our example, calculating \( \sec 75^{\circ} \) involved determining the complement \( 90^{\circ} - 75^{\circ} = 15^{\circ} \), leading us to use the identity \( \sec 75^{\circ} = \csc 15^{\circ} \). Understanding complementary angles also helps in visualizing trigonometric functions on a unit circle, where the concept of complementarity often ties into symmetrical properties of these functions.

Sine and Cosecant Calculation

To find the value of \( \csc 15^{\circ} \), one essential step involves calculating \( \sin 15^{\circ} \). This is because the cosecant function is the reciprocal of the sine function:

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)

Finding \( \sin 15^{\circ} \) can involve using angle subtraction identities, like \( \sin(15^{\circ}) = \sin(45^{\circ} - 30^{\circ}) \). Using known values of common angles, such as \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), \( \sin 30^{\circ} = \frac{1}{2} \), and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), allows us to break down \( \sin(15^{\circ}) \) into calculable parts.Once you derive the exact value: \( \frac{\sqrt{6} - \sqrt{2}}{4} \), you take its reciprocal to calculate \( \csc 15^{\circ} \). With the reciprocal property:

• \( \csc(15^{\circ}) = \frac{4}{\sqrt{6} - \sqrt{2}} \)

Use a calculator to find its approximate value \( \approx 3.8637 \), which is expressed to four decimal places.