Problem 22 - Algebra 2 and Trigonometry

Question

In \(21-24,\) find, to the nearest tenth, the degree measures of all \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that make the equation true. $$ 5 \sin \theta-1=1-2 \sin \theta $$

Step-by-Step Solution

Answer

\( \theta \approx 16.6^{\circ} \) and \( \theta \approx 163.4^{\circ} \).

Step 1: Understand the problem

The problem involves solving for \( \theta \) in the equation \( 5 \sin \theta - 1 = 1 - 2 \sin \theta \) within the interval \( 0^{\circ} \leq \theta < 360^{\circ} \). Our goal is to find the values of \( \theta \) to the nearest tenth of a degree.

Step 2: Combine like terms

Rewrite the equation by moving all terms involving sine to one side, and constant terms to the other side: \( 5 \sin \theta + 2 \sin \theta = 1 + 1 \). This simplifies to \( 7 \sin \theta = 2 \).

Step 3: Solve for sine of theta

Divide both sides by 7 to isolate \( \sin \theta \): \( \sin \theta = \frac{2}{7} \).

Step 4: Calculate reference angle

Find the reference angle by taking the inverse sine: \( \theta_{ref} = \sin^{-1}\left(\frac{2}{7}\right) \approx 16.6^{\circ} \). This angle represents the closest value, assuming \( \theta \) is in the first quadrant.

Step 5: Determine all possible angle solutions

Since \( \sin \theta \) is positive, \( \theta \) could be in either the first or second quadrant. **First Quadrant:**\( \theta = \theta_{ref} = 16.6^{\circ} \)**Second Quadrant:**\( \theta = 180^{\circ} - \theta_{ref} = 180^{\circ} - 16.6^{\circ} = 163.4^{\circ} \)

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It is often denoted as \( \sin \theta \), where \( \theta \) is an angle, typically measured in degrees or radians. Sine is particularly useful in understanding relationships in right triangles and unit circles.

• For a given angle \( \theta \), \( \sin \theta \) represents the ratio of the length of the side opposite \( \theta \) to the hypotenuse in a right triangle.
• On the unit circle, \( \sin \theta \) corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle.
• The range of the sine function is from -1 to 1, which means \( \sin \theta \) will always fall within this interval.

This property is crucial when solving trigonometric equations, as it guides us to feasible solutions during our calculations.

Quadrants in Trigonometry

Understanding quadrants is essential when working with angles in trigonometry, as they help determine the sign and orientation of trigonometric functions like sine. The coordinate plane is divided into four quadrants:

• First Quadrant: Both x and y coordinates are positive. As a result, sine is positive here.
• Second Quadrant: x is negative, and y is positive. Sine remains positive because y is positive.
• Third Quadrant: Both x and y coordinates are negative, making sine negative since y is negative.
• Fourth Quadrant: x is positive, and y is negative. Sine is negative in this quadrant.

When we solve equations like \( \sin \theta = \frac{2}{7} \), the positive value of sine tells us that \( \theta \) could be in the first or second quadrant, guiding us to seek specific solutions within those areas.

Degree Measure

Degree measure is one of the primary units for expressing angles in trigonometry. Each full rotation around a circle is 360 degrees. Using degree measure, we can express angles within a straightforward and familiar system for calculations and problem-solving.

• An angle of \( 90^{\circ} \) represents a quarter turn, forming a right angle.
• \( 180^{\circ} \) is a half rotation, an important benchmark often used to transition between quadrants.
• To express angles precisely, we often round to the nearest tenth of a degree, as done in the given problem.

Degree measure is especially helpful when calculating the precise position of angles within specific intervals, making it easier to find suitable solutions for trigonometric equations.