Problem 44 - Algebra 2

Question

Use identities to write each equation in terms of the single angle \(\theta .\) Then solve the equation for \(0 \leq \theta<2 \pi .\) $$ \sin 2 \theta \sin \theta=\cos \theta $$

Step-by-Step Solution

Answer

The solutions to the equation within the interval \( 0 \leq \theta < 2\pi \) are \( \theta = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4, \pi/2, 3\pi/2 \).

Step 1: Use identity to rewrite equation

We can re-write the given equation by replacing \( \sin 2\theta \) with the identity \( 2\sin\theta \cos\theta \). This gives: \( 2\sin\theta \cos\theta \sin\theta = \cos\theta \).

Step 2: Reduce Equation

We can now simplify the equation, by grouping similar terms together. This gives: \( 2\sin^2\theta \cos\theta - \cos\theta = 0 \). We then factor out \( \cos\theta \) to get: \( \cos\theta(2\sin^2\theta - 1) = 0 \).

Step 3: Solve for \( \theta \)

The expression equals to zero implies either \( \cos\theta = 0 \) or \( 2\sin^2\theta - 1 = 0 \), which gives \( \sin\theta = \pm \sqrt{1/2} \). From the unit circle, \( \cos\theta = 0 \) when \( \theta=\pi/2, 3\pi/2 \) and \( \sin\theta=\sqrt{1/2} \) or \( \sin\theta=-\sqrt{1/2} \) when \( \theta=\pi/4, 3\pi/4, 5\pi/4, 7\pi/4 \).

Step 4: Check solutions

We substitute the solutions in the original equation to verify them. The successful solutions by substitution are \( \theta = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4, \pi/2, 3\pi/2 \).

Key Concepts

Trigonometric Equations

Trigonometric equations often involve functions such as sine, cosine, and tangent, and they require solving for an unknown angle within these functions. The goal is to find angle values, usually denoted by \( \theta \), that satisfy the equation. To solve these equations, we often use trigonometric identities, which are formulas that express one trigonometric function in terms of other trigonometric functions. In the exercise, we converted \( \sin 2\theta \) into \( 2\sin\theta \cos\theta \) using the double angle identity, which helped simplify the equation. This technique of rewriting using identities is crucial for solving complex trigonometric equations because it reduces the equation to an easier form that reveals insights into the solution.

When solving trigonometric equations, consider the interval over which the solution is needed, often \( 0 \leq \theta < 2\pi \), which corresponds to a full rotation around the unit circle. This understanding helps to determine all possible angle solutions within the given range.

Unit Circle

The unit circle is an essential tool for understanding trigonometric functions and solving equations. It is a circle with a radius of one, centered at the origin of a coordinate plane. The unit circle provides a geometric approach to understanding the values of sine, cosine, and tangent for angles between 0 and \( 2\pi \).

Each position on the unit circle corresponds to an angle \( \theta \) measured counter-clockwise from the positive x-axis, and the coordinates of each point on the circle represent the cosine and sine of that angle. Specifically, the x-coordinate gives \( \cos\theta \) and the y-coordinate gives \( \sin\theta \). For instance, the angle \( \theta = \pi/4 \) corresponds to the coordinates \( (\sqrt{2}/2, \sqrt{2}/2) \), showing that both sine and cosine equal \( \sqrt{2}/2 \) at this angle.

Using the unit circle, we can easily derive important information, such as when \( \cos\theta = 0 \), which occurs at \( \theta = \pi/2 \) and \( \theta = 3\pi/2 \) within a single rotation (\( 0 \leq \theta < 2\pi \)). The unit circle simplifies finding angle solutions by mapping trigonometric function values geometrically.

Angle Solutions

Finding angle solutions within a given range usually involves understanding how trigonometric functions behave on the unit circle. For the exercise, we needed to find \( \theta \) values that satisfy \( \cos\theta = 0 \) and \( \sin\theta = \pm \sqrt{1/2} \). These specific values point us to distinct positions around the unit circle.

Angles where \( \cos\theta = 0 \) happen at the vertical positions of the unit circle: \( \theta = \pi/2 \) and \( \theta = 3\pi/2 \). Angles where \( \sin\theta = \pm \sqrt{1/2} \) refer to angles on the diagonal lines of the circle, such as \( \theta = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4 \). Each angle is important as it reflects symmetry principles within the circle and helps in quickly identifying all solutions.

Understanding angle solutions requires comfort with both the algebraic manipulation of trigonometric identities and the geometric intuition derived from the unit circle. Combining these approaches allows for the comprehensive solving of trigonometric equations.