Problem 81 - Precalculus

Question

Find the smallest positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\tan \theta=-0.7813\) and the terminal side of \(\theta\) lies in quadrant IV.

Step-by-Step Solution

Answer

\(\theta \approx 321^\circ\)

Step 1: Understanding the Problem

We need to find the smallest positive angle \(\theta\) such that \(\tan \theta=-0.7813\), with \(\theta\) located in quadrant IV. In quadrant IV, the tangent function is negative.

Step 2: Calculating Reference Angle

First, find the reference angle by taking the arctangent of the absolute value. Calculate \(\theta_r = \arctan(0.7813)\) using a calculator in degree mode. \(\theta_r \approx 38.5275\) degrees.

Step 3: Adjust for Quadrant IV

Since \(\theta\) is in quadrant IV, and tangent is negative in this quadrant, \(\theta\) will be \(360^\circ - \theta_r\). This gives \(\theta = 360^\circ - 38.5275^\circ \).

Step 4: Final Calculation and Rounding

Perform the subtraction to find \(\theta\): \(\theta = 321.4725^\circ\). Round to the nearest degree to get \(\theta \approx 321^\circ\).

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. It relates the angles of a right triangle to the ratio of the opposite side to the adjacent side. Specifically, for an angle \( \theta \), the tangent function is defined as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

The tangent function has a unique property: it can take both positive and negative values, depending on the angle and its position in the coordinate plane.
In the context of the exercise, you are given an angle \( \theta \) such that \( \tan \theta = -0.7813 \). This tells you that the angle is positioned such that its tangent is negative.
This information is crucial for determining the possible quadrants in which \( \theta \) might lie. Since tangent is positive in quadrants I and III and negative in quadrants II and IV, our given problem restricts \( \theta \) to quadrants II and IV. However, the problem specifies quadrant IV.

Quadrant IV

In the Cartesian coordinate system, Quadrant IV is the lower right section of the plane. A key feature of angles in this quadrant is that their range falls between 270° and 360°.
In Quadrant IV, different trigonometric functions have distinct signs:

• Sine is negative
• Cosine is positive
• Tangent is negative

Given the function \( \tan \theta = -0.7813 \), this aligns perfectly with our earlier understanding that \( \theta \) must lie in a quadrant where tangent is negative.
The problem further asks for the smallest positive measure of \( \theta \) with its terminal side in Quadrant IV. This is determined by calculating the reference angle and then modifying it to reflect its position in Quadrant IV.

Reference Angle

A reference angle is the acute angle (less than 90°) that a given angle makes with the x-axis. It is always positive and is commonly used to simplify problems involving trigonometric functions.
To compute the reference angle \( \theta_r \), use the equation:

• \( \theta_r = \arctan(0.7813) \)

Here, we take the arctangent of the absolute value because reference angles are always positive.
By doing so, \( \theta_r \approx 38.5275 \) degrees. This reference angle is crucial for understanding the angle's true position and measure.
To find the angle in Quadrant IV, you step back from a full circle by subtracting the reference angle from 360°. This gives:

• \( \theta = 360° - \theta_r \approx 360° - 38.5275° \)
• \( \theta \approx 321° \)

Thus, the smallest positive measure for \( \theta \) that satisfies \( \tan \theta = -0.7813 \) in Quadrant IV is approximately 321°.