Problem 9 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \theta = -60^\circ \) in degrees and \( \theta = -\frac{\pi}{3} \) in radians.

Step 1: Understand the Exercise

We need to find the angle \( \theta \) given by \( \theta = \arctan(-\sqrt{3}) \). We need to express \( \theta \) both in degrees and in radians.

Step 2: Recall Arctan Function Properties

The \( \arctan(x) \) function, also known as the inverse tangent function, returns an angle whose tangent is \( x \). The range of \( \arctan \) is \((-\pi/2, \pi/2)\) in radians or \((-90^\circ, 90^\circ)\) in degrees.

Step 3: Determine Reference Angle

Since \( \tan(\theta) = -\sqrt{3} \), we recall that \( \tan(60^\circ) = \sqrt{3} \) or \( \tan(\pi/3) = \sqrt{3} \). Thus, \( \theta \) becomes negative to match the negative tangent value.

Step 4: Find \( \theta \) in Degrees

The reference angle is \( 60^\circ \). Therefore, in the range \((-90^\circ, 90^\circ)\), \( \theta = -60^\circ \).

Step 5: Convert Degrees to Radians

To convert \( -60^\circ \) to radians, use the conversion: \( -60^\circ \times \frac{\pi}{180^\circ} = -\frac{\pi}{3} \).

Step 6: Write Final Answers

The angle \( \theta \) in degrees is \( -60^\circ \) and in radians is \( -\frac{\pi}{3} \).

Key Concepts

Arctan Function

The arctan function, or inverse tangent, is a great tool for finding angles when you know the tangent's value. Here's how it works: given a number, the arctan function tells you the angle that corresponds to that value of tangent.

It's crucial to understand that the range for the arctan function is limited. This means the results you get for angles are always confined between (-90^\circ, 90^\circ) or (-\pi/2, \pi/2) in radians. This helps determine which quadrant the angle is in.

When the tangent value is negative, as in this example with \(-\sqrt{3}\), it indicates the angle will be in either the fourth quadrant (since tangent is positive in the first quadrant and negative in the fourth).

• The reference angle for \(\tan(\theta) = \sqrt{3}\) is \(60^\circ\) because \(\tan(60^\circ) = \sqrt{3}\).
• So, \(\theta\) becomes \(-60^\circ\).

This negative sign accounts for the angle being in the correct quadrant.

Degree-Radian Conversion

Converting between degrees and radians is a common task in trigonometry, and knowing the formula is essential. The key relationship is that 180 degrees is equivalent to \(\pi\) radians. This translates into the conversion formula: 1 degree = \(\pi/180\) radians.

Let's walk through the process of converting \(-60^\circ\) degrees to radians:

• Start with the degree value: \(-60^\circ\).
• Multiply \(-60^\circ\) by \(\pi/180\).
• Simplify to get the radian measure: \(-\pi/3\).

The formula provides a straightforward way to switch back and forth between these units, enhancing flexibility when solving trigonometric problems.

Tangent of an Angle

The tangent of an angle is a fundamental trigonometric function. It's defined as the ratio of the length of the opposite side to the adjacent side in a right-angled triangle. In terms of a unit circle, it's more easily grasped as the y-coordinate divided by the x-coordinate of the point where the terminal side of the angle intersects the circle.

One essential property to remember is that the tangent function repeats every \(180^\circ\) or \(\pi\) radians, making it periodic. This property can help simplify evaluation of tangent at angles greater than 90 degrees or \(\pi/2\) radians.

• For example, \(\tan(60^\circ) = \sqrt{3}\), which is essential for interpreting the arctan function when a calculator or tables are not available.

Understanding this foundation is useful for linking angles to their respective tangent values, especially when working with inverse functions like arctan.