Problem 36 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \sin 2 \theta+2 \sin \theta-2 \cos \theta=2 $$

Step-by-Step Solution

Answer

The solutions are \( \theta = 90^\circ + k\cdot360^\circ \), \( \theta = 270^\circ + k\cdot360^\circ \), and \( \theta = 180^\circ + k\cdot360^\circ \), where \(k\) is an integer.

Step 1: Equation Rearrangement

Begin by moving all terms to one side of the equation: \[ \sin(2\theta) + 2\sin\theta - 2\cos\theta - 2 = 0 \] This helps prepare the equation for solving by breaking it down.

Step 2: Use Double Angle Identity

Recall that the double angle identity for sine is \( \sin(2\theta) = 2\sin\theta\cos\theta \). Substituting this, the equation becomes: \[ 2\sin\theta\cos\theta + 2\sin\theta - 2\cos\theta - 2 = 0 \] This allows us to express the equation in terms of sine and cosine.

Step 3: Factorization

Factor out common terms if possible. Notice that there is a common factor of 2 present:\[ 2(\sin\theta\cos\theta + \sin\theta - \cos\theta - 1) = 0 \]This simplifies the equation.

Step 4: Further Simplification

Simplify the expression inside the parenthesis: \[ \sin\theta(\cos\theta + 1) - (\cos\theta + 1) = 0 \]

Step 5: Apply Zero Product Property

Use the zero product property. For the equation to be true, either term in the product must be zero: - \(\sin\theta = 1\) or \(-1\)- \(\cos\theta = -1 \) Thus, the equation: \[ \theta = 90^\circ + k\cdot360^\circ \, (\text{for } \sin\theta = 1) \]\[ \theta = 270^\circ + k\cdot360^\circ \, (\text{for } \sin\theta = -1) \] \[ \theta = 180^\circ + k\cdot360^\circ \, (\text{for } \cos\theta = -1) \] Where \( k \) is an integer.

Key Concepts

Double Angle Identity

The double angle identity is a useful tool in trigonometry that helps to simplify expressions involving angles doubled in size. In the given problem, we encounter the sine double angle identity: \( \sin(2\theta) = 2\sin\theta\cos\theta \). This identity is instrumental in transforming equations that involve \( \sin(2\theta) \) into terms involving \( \sin \theta \) and \( \cos \theta \).Using the double angle identity, complex trigonometric expressions can be rewritten, making them easier to solve. In our exercise, the identity allows us to replace \( \sin(2\theta) \) with \( 2\sin\theta\cos\theta \) in the equation. This substitution transforms the initial trigonometric equation into a polynomial form that opens the door for factoring, which is the next step in solving the problem.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions which relate the angles of a triangle to the lengths of its sides. In trigonometry, they are crucial in solving various kinds of equations and problems.In this exercise, both \( \sin \theta \) and \( \cos \theta \) appear in the transformed equation. By rewriting the equation: \( 2\sin\theta\cos\theta + 2\sin\theta - 2\cos\theta - 2 = 0 \), we begin to see how the sine and cosine functions can be organized and manipulated to simplify.- Because \( \sin\theta \) and \( \cos\theta \) are the basic building blocks, it's helpful to understand their unique properties such as periodicity and range, which are from \(-1\) to \(1\). This knowledge aids in predicting and calculating the potential values of \( \theta \) that would satisfy the equation.Understanding these two functions allows us to elegantly factorize and tackle the problem further.

Zero Product Property

The zero product property is a fundamental algebraic principle that allows us to solve equations structured as products set to zero. It states that if a product of multiple factors equals zero, then at least one of the factors must be zero.In the context of our problem, after using the double angle identity and reorganizing the terms, the equation is factored as follows: \( 2(\sin\theta(\cos\theta + 1) - (\cos\theta + 1)) = 0 \). By applying the zero product property, we set each factor inside the parentheses to zero:

• \( \sin\theta(\cos\theta + 1) = 0 \)
• \( -(\cos\theta + 1) = 0 \)

By solving these individual equations, we derive solutions for \( \theta \):- \( \sin\theta = 1 \) gives solutions like \( 90^\circ + k\cdot360^\circ \), where \( k \) is an integer.- \( \sin\theta = -1 \) leads to solutions such as \( 270^\circ + k\cdot360^\circ \).- \( \cos\theta = -1 \) provides solutions like \( 180^\circ + k\cdot360^\circ \).The zero product property simplifies the process of finding the roots and identifying all potential solutions to the equation.