Problem 14 - Algebra 2

Question

Find each angle measure to the nearest tenth of a degree. \(\cos ^{-1} 0.992\)

Step-by-Step Solution

Answer

The angle is approximately 7.2 degrees when rounded to the nearest tenth.

Step 1: Understanding inverse cosine

The inverse cosine function, \( \cos^{-1}\), returns an angle between 0 and 180 degrees (inclusive) whose cosine is a given number. In this case, the given number is 0.992.

Step 2: Evaluating the expression

The angle is found by evaluating \( \cos^{-1}(0.992)\). Most calculators have a function for inverse cosine, which can be used to make this calculation.

Step 3: Rounding to the nearest tenth

The result from step 2 is then rounded to the nearest tenth of a degree, as the problem asks to find the measurement to the nearest tenth of a degree.

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in mathematics, particularly when dealing with trigonometric functions like cosine. Angles are measured in degrees, a unit that represents the amount of turn between two straight lines that have a common end point known as the vertex. A full circle is divided into 360 degrees.
Understanding angle measurement is crucial when using the inverse cosine, or \(\cos^{-1}\), function. This function returns the angle whose cosine is a given number. By definition, when using inverse trigonometric functions like \(\cos^{-1}\), the angles returned are within specific ranges. For inverse cosine, the range is from 0 degrees to 180 degrees, encompassing all possible angles for cosine values that range from -1 to 1 inclusive. In practical calculations, the angle measurement gives us a clear and precise way to describe the position of a line or a vector in a plane.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They play a crucial role in fields such as geometry, physics, engineering, and many areas of mathematics. The basic trigonometric functions are sine, cosine, and tangent, each of which can be represented as a ratio of sides in a right-angled triangle.
The cosine function specifically relates the adjacent side to the hypotenuse of a right-angle triangle. The inverse cosine function, denoted as \(\cos^{-1}\), helps find the angle from the known ratio of these sides. It essentially reverses the cosine operation, providing an angle for a given cosine value. This operation is integral when solving problems involving angles, where a given ratio of sides is provided, and the angle needs to be determined. Calculators and certain mathematical software can compute these inverse functions, simplifying the otherwise complex calculation process.

Rounding to Nearest Tenth

Rounding numbers is a mathematical technique used to simplify numbers, making them easier to work with while ensuring the results remain as accurate as needed. Specifically, rounding to the nearest tenth refers to taking a number and adjusting it to one digit after the decimal point.
Here’s a quick way to round a number to the nearest tenth:

• Identify the digit in the tenths place. This is the first digit to the right of the decimal point.
• Look at the digit in the hundredths place, which is the second digit to the right of the decimal point.
• If the hundredths digit is 5 or greater, increase the tenths digit by 1.
• If the hundredths digit is less than 5, leave the tenths digit unchanged.

Applying this method, we can ensure that calculations involving angle measurements, such as those found using inverse trigonometric functions, are presented in a precise, yet manageable form, retaining significant precision for practical use while also making them digestible.