Problem 15 - Algebra and Trigonometry

Question

\(11-22\) . Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. \(\cos ^{-1}(-0.92761)\)

Step-by-Step Solution

Answer

The value is approximately 2.76109.

Step 1: Understanding the Problem

We are tasked with evaluating the expression \( \cos^{-1}(-0.92761) \) and finding its approximate value using a calculator, correct to five decimal places.

Step 2: Definition and Domain of \( \cos^{-1} \)

The expression \( \cos^{-1}(x) \) gives the angle whose cosine is \( x \). It is defined for \(-1 \leq x \leq 1\) and the output of \( \cos^{-1} \) is between \( 0 \) and \( \pi \). Since \(-0.92761\) is within the domain, \( \cos^{-1}(-0.92761) \) is defined.

Step 3: Using a Calculator

Input \( -0.92761 \) into a calculator with the inverse cosine function to find \( \cos^{-1}(-0.92761) \). Ensure that the calculator is set to radians unless otherwise specified by the context of the problem.

Step 4: Rounding to Five Decimal Places

After computing the value, round the result to five decimal places for precision, as required by the problem.

Step 5: Verification of the Calculation

Check that the rounded result indeed fits within the expected range \( 0 \leq \text{result} \leq \pi \), to ensure the value makes sense in the inverse cosine context.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the sides and angles of triangles. One fundamental aspect of trigonometry is the study of trigonometric functions, including sine, cosine, and tangent. These functions help us express and solve problems involving angles and distances.

• Sine, Cosine, and Tangent: The cosine of an angle in a right triangle, written as \( \cos(\theta) \), is the ratio of the adjacent side to the hypotenuse.
• Inverse Functions: The inverse cosine function, written as \( \cos^{-1}(x) \), allows us to find the angle whose cosine is \( x \).

Understanding trigonometry is essential not just in mathematics, but also in fields like physics, engineering, and even computer graphics, where it helps model and compute angles and distances.

Domain and Range

When dealing with functions like inverse cosine, knowing the domain and range is crucial. The domain refers to the set of all possible inputs, while the range refers to all possible outputs.

• Domain of \( \cos^{-1}(x) \): The input, \( x \), must satisfy \(-1 \leq x \leq 1\). This is because the cosine function inherently produces values in this range.
• Range of \( \cos^{-1}(x) \): The output, often expressed as \( \theta \), will lie between \(0\) and \(\pi\) radians. This provides a unique angle solution.

These restrictions ensure that the inverse cosine function is well-defined and provides a specific angle measurement for any valid input within this interval.

Calculator Usage

Calculators are invaluable tools when working with trigonometric functions like \( \cos^{-1}(x) \). Here's how to effectively use one to compute the value:

• Settings: Ensure your calculator is set to the correct mode, usually radians, unless specified otherwise. This is important since the range of \( \cos^{-1}(x) \) is commonly expressed in radians.
• Inputting Values: Carefully input the value, such as \(-0.92761\), using the inverse cosine function button, frequently labeled as 'acos' or sometimes \( \cos^{-1} \).
• Rounding: After obtaining the result, round the figure to the required number of decimal places, often five, to meet precision requirements.

Practicing these steps ensures accuracy in your calculations and helps confirm the result within the expected range of the function.

Angle Measurement

Angles can be measured in various units, with degrees and radians being the most common. Understanding these helps in interpreting and calculating results from trigonometric functions.

• Radians: This is the standard unit of angular measurement in mathematics. One complete circle equals \(2\pi\) radians.
• Degrees: Another popular unit where one complete circle is \(360\) degrees. Often used in non-mathematical contexts.
• Conversion: Converting between radians and degrees can be essential. Use \(180\) degrees equals \(\pi\) radians. So, \(1\) radian equals approximately \(57.2958\) degrees.

For instance, when \( \cos^{-1}(x) \) outputs an angle in radians, knowing these conversions helps avoid confusion and ensures proper interpretation in practical scenarios.