Problem 43 - Algebra 2

Question

Use identities to write each equation in terms of the single angle \(\theta .\) Then solve the equation for \(0 \leq \theta<2 \pi .\) $$ 2 \sin 2 \theta-3 \sin \theta=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( \theta = 0\) and \( \theta = \pi \).

Step 1: Rewrite the Equation Using the Double Angle Identity

Use the double angle identity \(2 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)\) to rewrite the equation as: \(2 \sin(\theta) \cos(\theta) - 3 \sin(\theta) = 0\)

Step 2: Factor Out the common factor

Factor out the common factor of \(\sin(\theta)\), resulting in: \(\sin(\theta) [2 \cos(\theta) - 3] = 0\)

Step 3: Set Each Factor Equal to Zero and Solve for \(\theta\)

Setting each factor to zero gives two equations: \(\sin(\theta) = 0\) and \(2 \cos(\theta) - 3 = 0\). Solving the first equation for \(\theta\) gives: \(\theta = 0\) , \(\pi\) (given that \(\theta\) is between 0 and \(2 \pi\)). Solving the second equation for \(\theta\) gives: \(\cos(\theta) = 3/2\). However, the value for \(\cos(\theta)\) is not possible as the range for cosine is [-1,1]. Therefore there are no solutions from the second equation.

Step 4: Recap the Value of \(\theta\)

Therefore, \(\Theta\) only takes two values following from our equation which are \(0\) and \(\pi\)

Key Concepts

Double Angle Identities

In trigonometry, double angle identities are crucial for simplifying equations involving angles twice as large as a given angle. The most common double angle identities apply to sine, cosine, and tangent functions. For example, the double angle identity for sine is:

• \( \sin(2 \theta) = 2 \sin(\theta) \cos(\theta) \)

This identity breaks down the function of a double angle into functions of a single angle \( \theta\). This simplification can be particularly helpful in solving trigonometric equations where the angles presented are different from the standard single angle format.
In our original problem, this identity was used to transform \(2 \sin(2 \theta)\) into \(2 \sin(\theta) \cos(\theta)\), allowing the equation to be expressed solely in terms of \(\theta\). This step is essential for simplifying the equation and making it possible to factor or solve.

Factoring Trigonometric Expressions

Factoring is an algebraic process applied to simplify trigonometric expressions. It involves identifying a common factor in each term and rewriting the expression with this factor outside a set of parentheses. This process reveals simpler components that might be solved individually.
In the example provided, after applying the double angle identity, the equation becomes \(2 \sin(\theta) \cos(\theta) - 3 \sin(\theta) = 0\). Here, \(\sin(\theta)\) is a common factor. Pulling \(\sin(\theta)\) out of the equation results in:

• \( \sin(\theta) [2 \cos(\theta) - 3] = 0 \)

This transformation makes it possible to split the equation into separate parts, which can be solved individually. Factoring allows students to handle complex expressions by breaking them into smaller, more manageable equations.
This method is effective because it usually reduces the difficulty of solving the original problem.

Solution of Equations with Trigonometric Functions

Solving equations with trigonometric functions often requires breaking down the equation using identities and factoring, followed by setting each factor equal to zero. Once an equation is broken down into factors, each term is solved separately.
In our problem:

• Solving \( \sin(\theta) = 0\) leads to solutions \( \theta = 0\) and \( \pi\), as these are the angles where the sine function equals zero within the range \(0 \leq \theta < 2\pi\).
• For the factor \(2 \cos(\theta) - 3 = 0\), solving for \(\cos(\theta)\) would give \(\cos(\theta) = \frac{3}{2}\). However, this solution is invalid because \(\cos(\theta)\) can only have values between -1 and 1.

The problem is approached by addressing each factor to determine possible values for \(\theta\).
By understanding that only solutions within the valid range for trigonometric functions work, it becomes clear why some factors might not offer valid solutions. This systematic approach ensures that the solutions derived are consistent with mathematical principles and the given domain of the function.