Problem 157 - Calculus Volume 1

Question

For the following exercises, solve the trigonometric equations on the interval \(0 \leq \theta<2 \pi.\) $$2 \tan ^{2} \theta=2$$

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\)

Step 1: Simplify the Equation

Start by dividing both sides of the equation \(2 \tan^2 \theta = 2\) by 2. This simplifies the equation to \(\tan^2 \theta = 1\).

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

To solve \(\tan^2 \theta = 1\), take the square root of both sides. This gives \(\tan \theta = \pm 1\).

Step 3: Determine the Angles for \(\tan \theta = 1\)

The tangent function equals 1 at specific angles. In the interval \(0 \leq \theta < 2\pi\), \(\tan \theta = 1\) for \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}\).

Step 4: Determine the Angles for \(\tan \theta = -1\)

Similarly, \(\tan \theta = -1\) at \(\theta = \frac{3\pi}{4}\) and \(\theta = \frac{7\pi}{4}\) within the given interval.

Key Concepts

Understanding the Tangent Function

The tangent function, denoted as \(\tan \theta\), is one of the basic trigonometric functions used to study angles and triangles. It is particularly interesting because it can take any real number as its value.
Unlike sine and cosine, which are bound between -1 and 1, the tangent function has no restrictions on its range. It measures the ratio of the opposite side to the adjacent side in a right-angled triangle. Its behavior is periodic, with a period of \(\pi\), meaning it repeats its values every \(\pi\) radians.
Knowing the behavior of the tangent function helps solve trigonometric equations, especially when angles and their properties are involved.

Finding Angle Solutions for Tangent

When solving trigonometric equations like \( \tan^2 \theta = 1 \), identifying the correct angle solutions is crucial. The key is recognizing where the tangent function hits specific values, like \(1\) or \(-1\).
For \(\tan \theta = 1\), there are key angle solutions in the given interval \(0 \leq \theta < 2\pi\).

• \(\theta = \frac{\pi}{4}\): This is a common angle where both sine and cosine are equal, resulting in a tangent value of 1.
• \(\theta = \frac{5\pi}{4}\): This angle is in the third quadrant, where both sine and cosine are negative, but their ratio (tangent) becomes positive 1 again.

Similarly, for \(\tan \theta = -1\), the angles are:

• \(\theta = \frac{3\pi}{4}\): Here, the sine is positive while the cosine is negative, yielding a negative tangent value.
• \(\theta = \frac{7\pi}{4}\): In this case, the roles are reversed with sine being negative and cosine positive, which again gives a -1 value for the tangent.

Understanding these values makes it easier to anticipate and calculate the angle solutions for equations involving tangent.

Intervals in Trigonometry

Trigonometry problems often specify intervals to narrow down possible solutions. This is crucial in defining the exact values of angles that fit the requirements of a trig function's cycle.
In the problem context, the interval is \(0 \leq \theta < 2\pi\). This interval covers one full cycle of the unit circle, ensuring that any angle we find falls within a typical period for trigonometric evaluations.
When searching for angle solutions, knowing an interval:

• Helps identify valid solutions that match the cycle of the function.
• Avoids confusion over repeating cycles and extraneous values.
• Ensures solutions are only counted once per interval, providing a clear and precise answer.

Mastering intervals in trigonometry allows for systematic approaches to solving trigonometric equations.