Problem 66 - 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. $$\tan \theta+1=0$$

Step-by-Step Solution

Answer

\( \theta = \frac{3\pi}{4}, \frac{7\pi}{4} \) or \( 135^{\circ}, 315^{\circ} \).

Step 1: Rearrange the Equation

We start with the equation \( \tan \theta + 1 = 0 \). To solve for \( \tan \theta \), we subtract \( 1 \) from both sides to get \( \tan \theta = -1 \).

Step 2: Solve for \( \theta \) in Radians

The tangent function is negative in the second and fourth quadrants. In radians, the smallest positive angles where \( \tan \theta = -1 \) are \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).

Step 3: Convert Radians to Degrees

To convert from radians to degrees, we use the conversion factor \( 180/\pi \). Thus, \( \theta = \frac{3\pi}{4} \) converts to \( \theta = 135^{\circ} \) and \( \theta = \frac{7\pi}{4} \) converts to \( \theta = 315^{\circ} \).

Step 4: Verify the Specific Solutions

Checking both solutions, at \( 135^{\circ} \), \( \tan 135^{\circ} = -1 \), and at \( 315^{\circ} \), \( \tan 315^{\circ} = -1 \). Thus, 135 degrees and 315 degrees are correct solutions.

Key Concepts

Radians

Radians are a unit of angular measure used in many areas of mathematics. One radian is the angle created when the radius of a circle is laid out along the circle's circumference. This gives a natural connection to the circle, where the circumference is \(2\pi\) times the radius. Hence, a full circle is \(2\pi\) radians.
To find angles in radians for trigonometric problems, you often rely on well-known angles like \( \frac{\pi}{4} \), \( \frac{\pi}{2} \), and others. These represent portions of the circle's circumference.

• \( \frac{\pi}{4} \) rad is 45 degrees.
• \( \frac{\pi}{2} \) rad is 90 degrees.

As in the exercise, knowing that \( \tan \theta = -1 \) at specific angles lets you pinpoint radians like \( \frac{3\pi}{4} \) and \( \frac{7\pi}{4} \). These precise values in radians let us work with trigonometric functions more easily.

Degrees

Degrees are a more common unit for measuring angles, especially outside of mathematical circles. The full circle is divided into 360 equal parts. Thus, one degree is \(1/360\) of a complete rotation.
The conversion between radians and degrees is crucial for switching between the two. The \(180 \degree\) corresponds to \(\pi\) radians. Therefore, converting radians to degrees involves multiplying by \( \frac{180}{\pi} \).
For example, converting \(\frac{3\pi}{4}\) radians to degrees: \[ \theta = \frac{3\pi}{4} \times \frac{180}{\pi} = 135\degree. \] This formula provides the necessary connection between radians and degrees, useful for those familiar with either system.

Tangent Function

The tangent function is one of the main trigonometric functions often written as \( \tan \theta \). It is defined as the ratio of the sine to the cosine of an angle. In a right triangle, this ratio can be visualized by the opposite side over the adjacent side.
In trigonometric circles, the tangent function can have infinite solutions due to periodicity. For \( \theta \), satisfying \( \tan \theta = -1 \) means the sine and cosine values have the same magnitude but are opposite in sign.
Knowing where \( \tan \theta = -1 \), specifically in the second quadrant ( \( \frac{3\pi}{4} \) radians) and fourth quadrant ( \( \frac{7\pi}{4} \) radians), helps find solutions across various angles.

Angle Measures

Understanding angle measures is fundamental in trigonometry. Angles describe the amount of turn between two lines or surfaces, and the way we express these turns (in radians or degrees) can vary depending on context.
In trigonometry, identifying angles where specific function values occur is essential. For instance, knowing which angles result in \( \tan \theta = -1 \) helps solve problems quickly.
Always strive to use the smallest positive angle measures, as these are typically more straightforward and universally recognized. For instance, rather than using negative angles, stick to equivalent positive measures like \( 315^{\circ} \) for more clarity.