Problem 19 - Algebra 2 and Trigonometry

Question

In \(15-20,\) find, to the nearest degree, the measure of an acute angle for which the given equation is true. $$ \csc \theta-1=3 \csc \theta-11 $$

Step-by-Step Solution

Answer

The measure of the acute angle is approximately 12 degrees.

Step 1: Simplify the Equation

Start by moving all terms involving \( \csc \theta \) to one side of the equation. The equation is \( \csc \theta - 1 = 3 \csc \theta - 11 \). Subtract \( \csc \theta \) from both sides: \(-1 = 2 \csc \theta - 11 \).

Step 2: Isolate \( \csc \theta \)

Add 11 to both sides to get \( 2 \csc \theta = 10 \). Then, divide both sides by 2 to solve for \( \csc \theta \): \( \csc \theta = 5 \).

Step 3: Find \( \sin \theta \)

Since \( \csc \theta = \frac{1}{\sin \theta} \), if \( \csc \theta = 5 \), then \( \sin \theta = \frac{1}{5} \).

Step 4: Calculate the Angle \( \theta \)

To find \( \theta \), use the inverse sine function. Compute \( \theta = \sin^{-1}\left(\frac{1}{5}\right) \).

Step 5: Round to the Nearest Degree

After calculating \( \theta \) using a calculator, find that \( \theta \approx 11.54^d\). Round this to the nearest degree to get \( 12^)\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics, especially for solving equations that involve angles. These functions help us navigate back from a trigonometric ratio to an actual angle measurement. In the case of this exercise, we are dealing with the cosecant function, which is the reciprocal of the sine function. This makes the inverse process necessary when finding the angle. By defining the equation where \( \csc \theta = 5 \), we can rearrange this to \( \sin \theta = \frac{1}{\csc \theta} = \frac{1}{5} \). Then, to determine the angle \( \theta \), we utilize the inverse sine function, denoted as \( \sin^{-1} \). In a calculator, entering \( \sin^{-1}\left(\frac{1}{5}\right) \) gives us the measurement for \( \theta \). This calculation hinges on knowing how to manipulate inverse trigonometric functions, allowing us to reverse-engineer the angle from a given sine value.

Solving Trigonometric Equations

Solving trigonometric equations often involves multiple steps and the clever manipulation of trigonometric identities. In the exercise, the equation starts as \( \csc \theta - 1 = 3 \csc \theta - 11 \), which initially seems complex. However, by applying basic algebraic principles, the problem simplifies substantially.

• Move terms involving \( \csc \theta \) to one side.
• Simplify the equation by isolating the variable \( \csc \theta \).
• Solve the basic algebraic equation to find a simpler form \( \csc \theta = 5 \).

These steps involve elementary algebra and the understanding of trigonometric identities, enabling you to extract a meaningful solution from a complicated equation. The simplicity achieved through careful rearrangement and simplification is at the heart of solving such equations, leading to a straightforward form that can easily be solved.

Acute Angle Measurement

Accurately determining an acute angle measurement requires understanding the trigonometric principles and careful calculation. An acute angle is one that lies between 0 and 90 degrees, which is the typical domain where we seek results in these types of problems.

• The inverse functions, especially \( \sin^{-1} \), are used to find angle \( \theta \).
• When the result is found using a calculator, the output might be a decimal.
• Rounding ensures that the angle remains accurate and interpretable in practical scenarios.

For instance, after solving "\( \sin \theta = \frac{1}{5} \), the calculator provides a more precise angle, such as \( 11.54 \) degrees.
Rounding to the nearest degree, we find the final angle \( \theta = 12 \) degrees.
Such calculations highlight not only the precision of trigonometric evaluation but also the practicality of rounding in achieving a useable result for everyday interpretation.