Problem 96 - Precalculus

Question

Find the measure of the smallest positive angle \(\theta\) (rounded to the nearest degree) if \(\cos \theta=-0.2388\) and the terminal side of \(\theta\) (in standard position) lies in quadrant III. Solution: Evaluate with a calculator. \(\theta=\cos ^{-1}(-0.2388)=103.8157^{\circ}\) Approximate to the nearest degree. \(\theta \approx 104^{\circ}\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

Angle in quadrant III is approximately \(256^\circ\).

Step 1: Understanding the Problem

We need to find the smallest positive angle \(\theta\) whose cosine value is \(-0.2388\) and lies in quadrant III. The answer should be expressed in degrees.

Step 2: Use the Inverse Cosine Function

First, calculate \(\theta\) using the inverse cosine function: \[ \theta = \cos^{-1}(-0.2388) \] Evaluating this expression using a calculator gives \(\theta = 103.8157^\circ\).

Step 3: Identify the Quadrant

The inverse cosine function gives a principle value in the range \([0^\circ, 180^\circ]\), but \(103.8157^\circ\) lies in quadrant II. We are looking for an angle in quadrant III.

Step 4: Find Correct Quadrant III Angle

To find the equivalent angle in quadrant III, use \(180^\circ + \theta'\), where \(\theta'\) is the reference angle calculated: \[ \theta' = 180^\circ - 103.8157^\circ = 76.1843^\circ \] Thus, the quadrant III angle is: \[ 180^\circ + 76.1843^\circ = 256.1843^\circ \]

Step 5: Round to the Nearest Degree

Finally, round \(256.1843^\circ\) to the nearest whole number: \(\theta \approx 256^\circ\). This is the smallest angle in standard position that meets the given conditions.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential for finding angles when given a trigonometric ratio. If you know the value of a trigonometric function like sine, cosine, or tangent, inverse functions such as \( \sin^{-1} \), \ \( \cos^{-1} \)\, and \ \( \tan^{-1} \)\ help you work backward to find the angle.

In our example, to find \( \theta \) when \ \( \cos \theta = -0.2388 \), we use the inverse cosine function. This function helps us find \( \theta \) within the principal range of [0° to 180°], which is typically for angles in quadrants I and II.

Remember, the result from an inverse trigonometric function is often not the final answer if specific conditions, like quadrant location, are given. To adjust for quadrant constraints, additional calculations or adjustments are necessary. In our example, the angle result from the inverse cosine function needed further adjustment to meet the condition that \( \theta \) is in quadrant III.

Quadrant Angles

Quadrant angles help us understand where an angle lies on the coordinate plane. The coordinate plane is divided into four quadrants: Quadrant I (0° to 90°), Quadrant II (90° to 180°), Quadrant III (180° to 270°), and Quadrant IV (270° to 360°).

When an angle is said to lie in, say, Quadrant III, it means it falls between 180° and 270°. In our problem, the goal was to find the angle whose cosine is -0.2388 and lies in Quadrant III. Direct use of the inverse cosine function gave us 103.8157°, which is in Quadrant II, not III.

To find the angle in Quadrant III, we used a reference angle approach. We evaluated the reference angle as \( \theta' = 180° - 103.8157° \), which results in 76.1843°. Then, adding this to 180° (since Quadrant III starts at 180°), we find \( \theta = 180° + 76.1843° = 256.1843° \), confirming the correct quadrant placement.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where the functions are defined. They play a crucial role in simplifying expressions and solving trigonometric equations.

While our specific problem focused on using inverse functions and quadrants to find an angle, identities like the Pythagorean identities or the angle sum and difference identities can also be instrumental in different contexts.

For cosine, the identity \( \cos(\theta) = -\cos(180° - \theta) \) clarifies why the principal value returned by the inverse cosine function is in the second quadrant. This negative relationship in cosine helps us deduce complementary and supplementary angles—crucial when determining which quadrant an angle truly resides in, as occurred when recalculating from a principal to a quadrant-specific angle. Don't forget, mastering identities empowers you to tackle a multitude of trigonometric challenges effectively.