Problem 22 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\). $$ \sqrt{2} \cos \theta-\sqrt{2}=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{2} \cos \theta - \sqrt{2} = 0\) for \(0 \leq \theta<2 \pi\) is \(\theta = 0\).

Step 1: Isolate cosine function

First, manipulate the equation to isolate the cosine function: \( \sqrt{2} \cos \theta-\sqrt{2}=0 \Rightarrow \cos \theta = 1\). This step is valid because we came from \(\sqrt{2} \times \cos (\theta) = \sqrt{2}\) and then divided by \(\sqrt{2}\) to get \(\cos (\theta) = 1\).

Step 2: Find theta values

Next, remember that cosine of an angle equals 1 if the angle is \(0\) or \(2\pi\) in the given interval \([0, 2\pi)\). Therefore, the valid values of \(\theta\) that solve the equation are \(0\) and \(2\pi\).

Step 3: Check that solutions are in interval

Make sure the solutions obtained in step 2 are within the given interval (i.e., \(0 \leq \theta<2 \pi\)). Both \(0\) and \(2\pi\) are indeed within this interval. However, since \(2\pi\) and \(0\) represent the same point on the unit circle, but the interval explicitly excludes \(2\pi\), the only correct solution in this interval is \(\theta = 0\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It relates an angle of a right triangle to the lengths of the sides of the triangle.
When you see \(\cos \theta\), it refers to the x-coordinate of a point on the unit circle corresponding to a given angle \(\theta\).
It ranges from \(-1\) to \(1\), meaning its outputs are always within this interval.

• \(\cos \theta = 1\) at an angle of \(0\) or \(2\pi\), which means the point is at the far right of the unit circle.
• \(\cos \theta = -1\) at \(\pi\), placing the point at the far left.
• \(\cos \theta = 0\) at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).

The cosine function is periodic, repeating every \(2\pi\). This property is crucial for solving trigonometric equations as it allows us to consider the angle's specific position within a given interval.

Unit Circle

The unit circle is a circle with a radius of \(1\) centered at the origin of the coordinate plane. It's an essential tool for understanding trigonometric functions because it provides a visual representation.
In the context of the cosine function, every angle \(\theta\) corresponds to a point \((\cos \theta, \sin \theta)\) on the unit circle.

• The angle is measured starting from the positive x-axis, moving counter-clockwise.
• At \(\theta = 0\) or \(2\pi\), the angle returns to the starting point \((1, 0)\).
• This point is significant as this is where \(\cos \theta = 1\), giving one of the solutions to our original equation.

Visualize the unit circle whenever you need to find angles that satisfy specific trigonometric equations. It helps to determine where the angle's values fit within the circle and their corresponding coordinates.

Interval Notation

Interval notation is a way of representing a set of numbers along a number line. It's useful for specifying the range of solutions for equations.
In our problem, the interval \([0, 2\pi)\) indicates that \(\theta\) can start from \(0\) and approach \(2\pi\) without actually including \(2\pi\).

• \([0, 2\pi)\) means \(0\) is included, but \(2\pi\) is not.
• It's important for determining valid solutions, especially in periodic functions like the cosine.
• In the context of the unit circle, \(2\pi\) and \(0\) are essentially the same point, thus \(\theta=0\) is the correct solution.

Remember that interval notation helps in specifying where the valid solutions to trigonometric equations lie, preventing confusion over which endpoints are included.