Problem 18 - Algebra 2 and Trigonometry

Question

In \(13-24,\) find, to the nearest ten-thousandth, the radian measure \(\theta\) of a first-quadrant angle with the given function value. $$ \cos \theta=0.8267 $$

Step-by-Step Solution

Answer

The radian measure is approximately 0.6049.

Step 1: Identify the Inverse Function

To find the angle \( \theta \) when \( \cos \theta = 0.8267 \), we need to use the inverse cosine function. The inverse cosine is written as \( \theta = \cos^{-1}(x) \), and it returns the angle in radians when the cosine of that angle equals \( x \).

Step 2: Calculate the Angle Using Inverse Cosine

Using a calculator, input \( \cos^{-1}(0.8267) \) to find \( \theta \). Make sure your calculator is set to radians mode.

Step 3: Determine the Radian Value

Perform the calculation: \[ \theta = \cos^{-1}(0.8267) \approx 0.6049 \] radians.

Step 4: Round to the Nearest Ten-Thousandth

The radian measure \( \theta \approx 0.6049 \) is already rounded to the nearest ten-thousandth.

Key Concepts

Understanding the Inverse Cosine Function

The inverse cosine function is a crucial tool in trigonometry, especially when you need to find the angle \( \theta \) given a cosine value. When you have a trigonometric equation like \( \cos \theta = 0.8267 \), you can use the inverse cosine, written as \( \theta = \cos^{-1}(x) \), to solve for \( \theta \).

• The inverse cosine function helps in finding the angle corresponding to a specific cosine value.
• It is denoted by \( \cos^{-1} \) or sometimes "arccos".
• It returns an angle in radians, which is the standard unit for angles in trigonometry.

Keep in mind, the value you input into \( \cos^{-1} \) must be within the range of the cosine function, which is from -1 to 1. This is why using a calculator correctly is vital—to ensure you're in the right mode (radians) when finding the radian measure of an angle.

Exploring First-Quadrant Angles

First-quadrant angles are a specific category of angles in trigonometry. They lie between 0 and \( \frac{\pi}{2} \) radians or 0° and 90°. When you calculate an angle using inverse functions like inverse cosine, you often end up with a first quadrant angle if the input value is positive.

• A first-quadrant angle is always positive.
• You will find these angles when dealing with standard trigonometric problems because they represent the simplest and most familiar part of the unit circle.
• In the unit circle, all trigonometric functions (sine, cosine, tangent) are positive in the first quadrant.

By focusing on the first quadrant, you can understand basic trigonometric relationships without the added complexity of negative coordinates or angles exceeding \( \frac{\pi}{2} \).

The Role of Trigonometric Functions

Trigonometric functions are at the heart of calculating angles and understanding geometric relationships. They include sine, cosine, tangent, and their respective inverses, each highlighting a different aspect of a triangle's dimensions or angles.

• Cosine, specifically, relates the angle to the adjacent side and hypotenuse in a right triangle or to the x-coordinate in a unit circle.
• These functions allow transformations between angle measures and side lengths, facilitating solutions for a variety of geometric problems.
• Inverse trigonometric functions like \( \cos^{-1} \) are key for back-calculating angles when given side ratios.

Using these functions effectively requires a good understanding of their ranges, behaviors, and relationships to angles within the unit circle. Mastery of trigonometric functions helps in solving both basic and complex mathematical problems related to angles.