Problem 17 - Algebra 2

Question

Find each angle measure to the nearest tenth of a degree. \(\cos ^{-1} \frac{3}{8}\)

Step-by-Step Solution

Answer

The angle measure to the nearest tenth of a degree of \(\cos^{-1}\frac{3}{8}\) is approximately 67.98 degrees.

Step 1: Analyze the problem

The problem statement requires finding the angle measure to the nearest tenth of degree for a given value \( \cos^{-1}\frac{3}{8} \). This is an arccosine function, which gives the angle whose cosine is 3/8.

Step 2: Find the arccosine

To find the angle measure, use a calculator to find the arccosine or inverse cosine of 3/8. Set your calculator to degree mode because the answer should be in degrees.

Step 3: Interpret the result

The result of this operation gives the exact measure of the angle in degrees. This is the angle whose cosine is 3/8.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that expresses the relationship between the angle and the adjacent side over the hypotenuse in a right triangle. For a given angle \(\theta\), the cosine is defined by the formula:
\[\cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\]The range of the cosine function is between -1 and 1, inclusive. It plays a critical role in calculating angles and distances in trigonometry.
One important aspect of cosine is its periodicity, meaning it repeats its values in a regular interval, specifically 360 degrees or \(2\pi\) radians. This property is essential for solving problems involving angles larger than 360 degrees.

• Cosine is even: \(\cos(-\theta) = \cos(\theta)\)
• Cosine decreases as the angle approaches 90 degrees.

Understanding these properties helps in working with its inverse, the arccosine function, used to determine angles from given cosine values.

Angle Measurement

Angle measurement is crucial in understanding and computing various geometric and trigonometric problems. Angles can be measured in different units, but the most common ones are degrees and radians.
- Degrees: One full circle is 360 degrees. The degree system is often used in everyday applications and is intuitive for visualizing angles.- Radians: This is the standard unit of angular measure in mathematics where one full circle is \(2\pi\) radians. It's required for many theoretical calculations in calculus and advanced mathematics.
When performing trigonometric calculations, it's important to know which unit system is being used. Each unit provides a different interpretation of angle size, so switching between them correctly ensures accurate results. Understanding angles this way allows us to work effectively with functions such as arc cosine, translating cosine values back into measurable angles.

Degree Mode

In trigonometry, setting your calculator to degree mode is essential when calculations require the result in degrees. Degree mode interprets the trigonometric functions based on a circle divided into 360 equal parts.
When calculating the arccosine, as in \(\cos^{-1}(\frac{3}{8})\), the calculator will return an angle provided that it's in degree mode. This mode converts numeric values directly to degrees, which are often required in geometrical problems and practical applications.

• Ensure correct mode for expected results.
• Degrees are intuitive for practical use.

Always remember to check and set your calculator to degree mode when solving problems that require answers in degrees, as mode errors lead to incorrect answers.

Trigonometric Calculations

Trigonometric calculations are techniques used to find unknown angles or sides of triangles using known values and trigonometric identities. Functions like sine, cosine, and tangent form the basis of these calculations. The arccosine function, \(\cos^{-1}\), specifically helps us determine the angle given the cosine value. Here’s how it works:
Suppose you need to find the angle \(\theta\) such that \(\cos\theta = \frac{3}{8}\). To find \(\theta\), you use the inverse cosine function \(\cos^{-1}(\frac{3}{8})\), which you compute using a calculator.
Key tips include:

• Use a calculator for precision.
• Double-check calculator mode settings.

Applying these rules ensures accurate and effective trigonometric calculations, allowing you to solve complex geometric problems with ease.