Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\frac{2 \tan \theta}{3-\tan ^{2} \theta}=1$$

Step-by-Step Solution

Answer

Angles: \( \theta = \frac{\pi}{4}, 1.8925 \) rad or \( 45^{\circ}, 108.4^{\circ} \).

Step 1: Understand the Equation

You're given the equation \( \frac{2 \tan \theta}{3-\tan^2 \theta} = 1 \). We need to find the values of \( \theta \) that satisfy this equation. First, let's simplify it.

Step 2: Simplify the Equation

Multiply both sides of the equation by \( 3 - \tan^2 \theta \), leading to \( 2 \tan \theta = 3 - \tan^2 \theta \). This simplifies to a quadratic form in \( \tan \theta \).

Step 3: Form a Quadratic Equation

Rearrange the equation to form a quadratic equation: \( \tan^2 \theta + 2 \tan \theta - 3 = 0 \). This is a standard quadratic equation in \( \tan \theta \).

Step 4: Solve the Quadratic Equation

Use the quadratic formula to find \( \tan \theta \):\[ \tan \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \( a = 1 \), \( b = 2 \), and \( c = -3 \).Substitute to get:\[ \tan \theta = \frac{-2 \pm \sqrt{2^2 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2} \]This gives two solutions: \( \tan \theta = 1 \) and \( \tan \theta = -3 \).

Step 5: Find Angles for \( \tan \theta = 1 \)

For \( \tan \theta = 1 \), the primary angle is \( \theta = \frac{\pi}{4} \) radians or \( 45^{\circ} \). It's positive and within the first cycle (0 to 2\( \pi \)).

Step 6: Find Angles for \( \tan \theta = -3 \)

For \( \tan \theta = -3 \), solve for \( \theta \) using the inverse tangent function: \( \theta = \arctan(-3) \). This will be a negative angle; adding \( \pi \) gives the corresponding positive angle in the range of 0 to 2\( \pi \). Calculate using a calculator to find \( \theta \approx -1.249 \) radians. Adding \( \pi \) gives \( \theta \approx 1.8925 \) radians or \( 108.4^{\circ} \).

Step 7: Determine Nonnegative Angles

Convert the found angles to the least possible nonnegative measure. The nonnegative angles in radians are \( \frac{\pi}{4} \) and approximately 1.8925 rad, and in degrees are \( 45^{\circ} \) and \( 108.4^{\circ} \).

Key Concepts

Radians

When working with angles, two units of measurement are commonly used: degrees and radians. Radians offer a natural way to quantify angles, especially in mathematics and physics. One full circle around a point is equivalent to an angle of 360 degrees or 2π radians.
This means half a circle, or a straight line, spans an angle of π radians, which is equal to 180 degrees.
Understanding how to convert between radians and degrees is helpful and often necessary:

• To convert from degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \).
• To convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \).

This conversion is crucial when solving trigonometric equations, as trigonometric functions often use radians in their calculations.
In trigonometry exercises, it’s common to encounter radians while determining solutions, as they provide precise and uncomplicated mathematical representations of angles.

Quadratic Equation

A quadratic equation is any equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable.
The characteristic of a quadratic is its highest exponent of 2 on its variable. When solving trigonometric problems, quadratics often appear, requiring methods such as factoring, completing the square, or using the quadratic formula.

For our exercise, we're dealing with the trigonometric identity \( \tan \theta \), but the process is similar to algebraic quadratics. First, identify \( a \), \( b \), and \( c \) from the equation \( \tan^2 \theta + 2 \tan \theta - 3 = 0 \):

• \( a = 1 \)
• \( b = 2 \)
• \( c = -3 \)

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is then applied to find the roots of \( \tan \theta \).
Calculating these roots provides the solution we need to determine the value of \( \theta \) in the context of the trigonometric problem. This approach demonstrates how algebraic techniques translate into trigonometric equations.

Inverse Tangent

The inverse tangent, or arctan, function is crucial when determining angles from trigonometric ratios. The function, denoted as \( \tan^{-1}x \) or \( \arctan x \), returns the angle whose tangent is \( x \).
This function produces angles typically in the range of \(-\frac{\pi}{2}\) to \( \frac{\pi}{2} \) radians. In solving the given trigonometric equation, inverse tangent helps retrieve angle values for non-standard tangents like \( \tan \theta = -3 \).

During calculations:

• The angle for \( \tan \theta = 1 \) is straightforward, resulting in \( \theta = \frac{\pi}{4} \).
• For \( \tan \theta = -3 \), \( \arctan(-3) \) results in a negative angle, illustrating the inverse tangent's typical range.
• Adjust these angles for a nonnegative angle measure by adding \( \pi \) when required, ensuring the angles are within 0 to \( 2\pi \).

This adjustment is essential to provide least possible nonnegative angle measures to comply with standard practice for angle determination in radians. This step ensures all results conform to expectations in mathematical and practical applications.