Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find each value of \(\theta : \mathbf{a} .\) in degrees \(\mathbf{b} .\) in radians $$ \theta=\arctan (-1) $$

Step-by-Step Solution

Answer

\( \theta = 135^\circ \) or \( -45^\circ \); in radians: \( \theta = \frac{3\pi}{4} \) or \( -\frac{\pi}{4} \).

Step 1: Understand the Function

The function we need to evaluate is \( \theta = \arctan(-1) \). The \( \arctan \) function, also known as the inverse tangent function, returns the angle whose tangent is \(-1\).

Step 2: Determine the Angle in Quadrant

Tangent is negative in the second and fourth quadrants. For \( \tan(\theta) = -1 \), \( \theta \) could either be \(135^\circ\) (second quadrant), or \(-45^\circ\) (fourth quadrant).

Step 3: Convert Degrees to Radians

To convert degrees to radians, use the formula \( \text{{radians}} = \text{{degrees}} \times \frac{\pi}{180} \). Therefore, \(-45^\circ\) is \( -45 \times \frac{\pi}{180} = -\frac{\pi}{4} \) radians.

Key Concepts

Understanding the Arctan Function

The arctan function, also called the inverse tangent function, is crucial in trigonometry. It allows us to find an angle whose tangent is a specific value. Unlike the basic tangent function, which takes an angle and returns a ratio, arctan takes a ratio and returns an angle. Here, the problem asks us to evaluate \( \theta = \arctan(-1) \), meaning we need to find the angle whose tangent is \(-1\).

The tangent of an angle can be seen as the ratio of the opposite side to the adjacent side in a right triangle. Thus, finding an angle with a tangent of \(-1\) means that the lengths of the opposite and adjacent sides are equal in magnitude but have opposite signs.

• The basic range of the arctan function is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians.
• This corresponds to angles between \(-90^\circ\) and \(90^\circ\).

To summarize, \(\arctan(-1)\) yields an angle of \(-\frac{\pi}{4}\) radians or \(-45^\circ\) in degrees within the standard range of values for the arctan function.

Angle Conversion: Degrees to Radians

Angle conversion is crucial because different contexts often require different units. Here, the problem gives us the angle in degrees \((-45^\circ)\) and asks us to convert it into radians. Knowing how to switch between degrees and radians is a key skill in trigonometry, especially in calculus and advanced mathematics.

The formula to convert an angle from degrees to radians is:\[ \text{{radians}} = \text{{degrees}} \times \frac{\pi}{180}\]So to convert \(-45^\circ\) to radians, multiply it by \(\frac{\pi}{180}\):

• \(-45 \times \frac{\pi}{180} = -\frac{\pi}{4}\)

By performing this conversion:

• \(-45^\circ\) becomes \(-\frac{\pi}{4}\) radians.

This conversion is often useful in computations where radian measure simplifies calculations, particularly in calculus.

Quadrants in Trigonometry

Understanding quadrants is essential when dealing with inverse trigonometric functions. Quadrants are divisions of the coordinate plane and help define the sign of trigonometric functions.

There are four quadrants in the Cartesian plane:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, sine is negative.

For \( \theta = \arctan(-1) \), we need to determine which quadrants could yield the tangent value of \(-1\). Tangent is negative in the second and fourth quadrants because it is the ratio of sine to cosine.

• In the second quadrant, the angle that gives tangents of \(-1\) is \(135^\circ\).
• In the fourth quadrant, the angle yielding the same tangent is \(-45^\circ\).

Choosing from these, \(-45^\circ\) is within the principal value range of the arctan function, which makes it the solution to the exercise where \(\theta = \arctan(-1)\). Understanding these quadrant properties helps in solving various trigonometric problems effectively.