Problem 14 - 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. $$ \cos \theta=0.5379 $$

Step-by-Step Solution

Answer

The radian measure \( \theta \) is approximately \( 0.9962 \).

Step 1: Understanding the Problem

We need to find the angle \( \theta \) in radians, located in the first quadrant, such that \( \cos \theta = 0.5379 \). We need to round the answer to the nearest ten-thousandth.

Step 2: Using the Inverse Trigonometric Function

Since \( \theta \) is in the first quadrant, the cosine function is positive. We use the inverse cosine function to find \( \theta \). We write: \( \theta = \cos^{-1}(0.5379) \).

Step 3: Calculation with a Calculator

Using a calculator, compute the inverse cosine of 0.5379. This calculation gives \( \theta \approx 0.996239 \) radians.

Step 4: Rounding the Result

We are to round \( \theta \approx 0.996239 \) radians to the nearest ten-thousandth place. The ten-thousandths digit is 2, and the following digit is 3, so we round down.

Step 5: Final Answer

After rounding, the radian measure of the angle \( \theta \) is approximately \( 0.9962 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are like the opposites of regular trigonometric functions. They tell us the angle that corresponds to a given trigonometric value. For example, if you know that the cosine of angle \( \theta \) is 0.5379, you can find \( \theta \) itself by using the inverse cosine function, written as \( \cos^{-1} \).

The primary inverse trigonometric functions are inverse sine (\( \sin^{-1} \)), inverse cosine (\( \cos^{-1} \)), and inverse tangent (\( \tan^{-1} \)). Each of these functions helps determine the angle with a specific sine, cosine, or tangent value, respectively.

• Inverse Sine: \( \sin^{-1} \) is used to find an angle whose sine value is known.
• Inverse Cosine: \( \cos^{-1} \) is used for finding an angle from a known cosine value.
• Inverse Tangent: \( \tan^{-1} \) helps to find an angle with a given tangent value.

Understanding inverse trigonometric functions is crucial for solving problems where you need to determine angles, like in the original exercise where \( \cos(0.5379) \) was used to find \( \theta \) in radians.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It relates an angle in a right triangle to the ratio of the length of the adjacent side to the hypotenuse.

For an angle \( \theta \), particularly in a unit circle, cosine represents the \( x \)-coordinate of the point where the terminal side of \( \theta \) intersects the circle.

• \( \cos(90^\circ) = 0 \)
• \( \cos(0^\circ) = 1 \)
• \( \cos(180^\circ) = -1 \)

The cosine function is periodic, repeating every \( 360^\circ \) (or \( 2\pi \) radians). Positive cosine values are generally found in the first and fourth quadrants. In the context of the original problem, the angle needed to have cosine equal to 0.5379 had to be in the first quadrant, where cosine values are indeed positive.

To find \( \theta \) using a given cosine value, as done in the exercise, the inverse cosine function \( \cos^{-1} \) is used.

First Quadrant Angles

In trigonometry, the unit circle is divided into four quadrants, each serving an important role in understanding angles and their trigonometric values. The first quadrant includes angles from \( 0 \) to \( \frac{\pi}{2} \) radians, or \( 0^\circ \) to \( 90^\circ \).

First quadrant angles are unique because all standard trigonometric functions—sine, cosine, and tangent—are positive here. This is why, in our exercise, when we needed an angle \( \theta \) for which \( \cos \theta = 0.5379 \), we derived that \( \theta \) lay in the first quadrant.

• Positive angles are measured counterclockwise from the positive \( x \)-axis.
• In the first quadrant, sine, cosine, and tangent of angles yield positive values.

Recognizing that \( \theta \) needed to be in the first quadrant was essential for correctly applying the inverse cosine function and ultimately determining the correct radian measure.