Problem 3 - Algebra 2

Question

Find all solutions of each equation for the given interval. \(\sin 2 \theta=\cos \theta ; 0 \leq \theta<2 \pi\)

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{\pi}{2}, \frac{3\pi}{2}\) in the interval [0, 2\pi).

Step 1: Use Double Angle Identity

Start by using the trigonometric identity for the double angle of sine: \(\sin 2\theta = 2\sin\theta \cos\theta\). Substitute this into the equation:\[ 2\sin\theta \cos\theta = \cos \theta \].

Step 2: Simplify the Equation

To solve the equation \(2\sin\theta \cos\theta = \cos \theta\), first, divide both sides by \(\cos \theta\) assuming \(\cos \theta eq 0\). This gives us \(2\sin\theta = 1\).

Step 3: Solve for \(\theta\) from Simplified Equation

Solve \(2\sin\theta = 1\) for \(\theta\). First, solve for \(\sin \theta\): \(\sin \theta = \frac{1}{2}\). This corresponds to angles \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\) within the interval \([0, 2\pi)\).

Step 4: Consider When \(\cos \theta = 0\)

Consider the case we ignored when dividing by \(\cos \theta\): \(\cos \theta = 0\). For \(\cos \theta = 0\), \(\theta\) corresponds to \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\) in the interval \([0, 2\pi)\).

Step 5: Compile All Solutions

Combine all found solutions. Thus, the solutions \(\theta\) where both equations \(2\sin\theta = 1\) and \(\cos \theta = 0\) are satisfied in the interval \([0, 2\pi)\) are: \(\theta = \frac{\pi}{6}\), \(\theta = \frac{5\pi}{6}\), \(\theta = \frac{\pi}{2}\), and \(\theta = \frac{3\pi}{2}\).

Key Concepts

Double Angle Identity

The Double Angle Identity is an essential tool in trigonometry, simplifying calculations that involve trigonometric functions with double angles. The double angle identity for sine is particularly useful. It is given by:
\[\sin 2\theta = 2\sin\theta \cos\theta\]This formula arises when a trigonometric expression has an angle double that of another, allowing us to reduce this to simpler single angle terms.
In our exercise, we used this identity at the very beginning to manipulate the given equation \(\sin 2\theta = \cos \theta\) into:

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

This step begins the simplification process, making the equation easier to solve for \(\theta\). By expressing a double angle in terms of the trigonometric functions of a single angle, we can work with more familiar and manageable relationships.

Interval Solutions

Interval solutions refer to solving equations within a specified range. In trigonometry, angles typically lie within a circumscribed interval, most commonly \([0, 2\pi)\), which represents a full rotation around a circle in radians.
For our exercise, finding solutions for the equation within this interval is crucial as the sine and cosine functions repeat their values in a predictable cycle.

• We first identify potential solutions by solving the simpler equations that were derived, such as \(2\sin\theta = 1\).
• Upon solving for \(\sin \theta = 0.5\), we find specific angles that satisfy this condition within the given range.

This results in solutions like \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\), which reflect where the sine function equals \(\frac{1}{2}\) when traversed through \([0, 2\pi)\).
Additionally, we must consider where the cosines equal zero, such as \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), to ensure all cases within the specified interval are covered.

Trigonometric Identities

Trigonometric identities play a significant role in simplifying expressions and solving equations in trigonometry. These identities are pre-established equalities that hold true for the trigonometric functions for all angles within their domain.
In the exercise, several identities are at play:

• Double Angle Identity: As discussed, it helps transform the angle \(2\theta\) into single angle terms \(\sin\theta\) and \(\cos\theta\).
• Sine and Cosine Values: Recognizing standard values of sine and cosine for known angles is critical. For instance, knowing \(\sin \theta = \frac{1}{2}\) at specific angles helps isolate solutions directly.

These identities are powerful because they allow us to break down complex trigonometric expressions, making it easier to find all possible solutions within any given interval such as \([0, 2\pi)\).
By understanding and applying these identities, solving trigonometric equations becomes more approachable, ensuring all potential solutions are identified.