Problem 11 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ \tan \theta+12=2 \tan \theta+11 $$

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{4}, \frac{5\pi}{4}\)

Step 1: Isolate the Trigonometric Function

Begin by moving all terms involving \(\tan \theta\) to one side of the equation. Subtract \(\tan \theta\) from both sides:\[ \tan \theta + 12 - \tan \theta = 2 \tan \theta + 11 - \tan \theta \]This simplifies to:\[ 12 = \tan \theta + 11 \]

Step 2: Solve for \(\tan \theta\)

Now isolate \(\tan \theta\) on one side by subtracting 11 from both sides:\[ 12 - 11 = \tan \theta \]This simplifies to:\[ \tan \theta = 1 \]

Step 3: Find Angles for the Tangent Value

The tangent function, \(\tan \theta\), equals 1 at certain angles within the given interval. Recall that \(\tan \theta = 1\) at:\[ \theta = \frac{\pi}{4}, \frac{5\pi}{4} \]These angles occur because the tangent function has a period of \(\pi\), meaning it repeats every \(\pi\) radians.

Step 4: Verify Intervals

Ensure the values found for \(\theta\) are within the specified interval \([0, 2\pi)\). Both \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}\) lie within the range 0 to \(2\pi\).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is one of the primary trigonometric functions. It is defined as the ratio of the sine of an angle to the cosine of that angle. In other words:\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]When solving trigonometric equations like the one in this exercise, you're often focusing on finding angles \( \theta \) for which the tangent function yields a specific value.

• Undefined Points: The function is undefined when \( \cos \theta = 0 \), which happens at \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \)
• Symmetry: The tangent function is odd, meaning that it mirrors through the origin: \( \tan(-\theta) = -\tan(\theta) \).

The tangent function turns out to be particularly useful in trigonometry because it relates directly to the geometry of right triangles and the unit circle. Understanding \( \tan \theta = 1 \) means looking at scenarios where this ratio reflects a specific angle or position on the unit circle.

Angle Measurement

In trigonometry, angles can be measured in degrees or radians. In this exercise, the interval is from \(0\) to \(2\pi\), which indicates that we are using radians.

• Radians: One full rotation around a circle is \(2\pi\) radians, equivalent to 360 degrees.
• Conversion: To convert between degrees and radians, you use the relation \(180^\circ = \pi \text{ radians}\).

Radian measurement is often preferred in mathematics, especially calculus, because it simplifies many formulas and calculations. For example, using radians means that the length of an arc of a unit circle is numerically equal to the angle in radians subtending the arc.For the tangent equation \( \tan \theta = 1 \), the angle solutions \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \) are determined with radians, showing where this specific tangent value occurs.

Periodicity of Trigonometric Functions

The periodicity of a trigonometric function refers to the function repeating its values at regular intervals. For the tangent function, this interval is \(\pi\). This characteristic is essential because it allows us to predict its behavior and solve equations over any given interval.

• Period of Tangent: For \( \tan \theta \), the function repeats every \( \pi \) radians.
• Repetition: If you have a solution \( \theta \), then \( \theta + n\pi \) (where \( n \) is any integer) is also a valid solution.

In the provided exercise, the solutions \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \) reflect this periodic nature. Since \( \tan \theta \) equals 1 every \( \pi \) radians, these angles were expected solutions within the interval from 0 to \(2\pi\). Understanding this periodicity is crucial for solving trigonometric equations, ensuring that you correctly identify all potential solutions within a specified range.