Problem 8 - Algebra and Trigonometry

Question

Use a calculator to find an approximate value of each expression rounded to five decimal places, if it is defined.. $$ \cos ^{-1}(-0.75) $$

Step-by-Step Solution

Answer

\(\cos^{-1}(-0.75) \approx 2.41886\) radians.

Step 1: Identify what is being asked

We are asked to find \(\cos^{-1}(-0.75)\). The \(\cos^{-1}\) function is the inverse of the cosine function, which means we need to find the angle whose cosine is -0.75.

Step 2: Check domain range of acos function

The range of \(\cos^{-1}\) is from \(0\) to \(\pi\), which means the function outputs angles in this interval. We must ensure that \(-0.75\) is within the domain of the \(\cos^{-1}\) function, where \(-1 \leq x \leq 1\). Since \(-0.75\) is within this domain, we can proceed.

Step 3: Compute the value using a calculator

Use a calculator to compute the value of \(\cos^{-1}(-0.75)\). Ensure that your calculator is set to work in radians. Enter \(-0.75\) and use the inverse cosine function (often labeled as \(\cos^{-1}\) or \(\text{acos}\)) to find the result.

Step 4: Round to five decimal places

After computing it on the calculator, suppose it shows an output like \(2.41885\), round the result to five decimal places to get a precise value, which is the final answer.

Key Concepts

Cosine Function

The cosine function is a fundamental aspect of trigonometry that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. This concept is crucial when working with right-angled triangles and also extends into the study of waves, oscillations, and rotations in calculus and physics.

In simple terms, if you have a right triangle, the cosine of an angle is:

• Cosine(\(\theta\)) = Adjacent side / Hypotenuse

The cosine function can be applied to any angle, not just those within a right triangle, through the unit circle. The unit circle allows the cosine function to be extended to all real numbers, measured in radians or degrees.

The cosine function has specific properties:

• It is a periodic function, repeating every 2\(\pi\) radians or 360 degrees.
• Its values range between -1 and 1, known as its range.
• At 0 and multiples of \(\pi\), the cosine function reaches its maximum value of 1 or its minimum value of -1.

Angle Measurement

Angles are a measure of rotation between two rays that originate from a common point, known as the vertex. In trigonometry, measuring angles is essential because it allows us to apply functions like sine, cosine, and tangent to solve problems.

Angles can be measured in several units:

• Degrees: A circle is divided into 360 equal parts, each part known as a degree. It is a more intuitive way for most people since many daily applications use degrees.
• Radians: A radian is based on the radius of a circle. It provides a more natural mathematical measure because it simplifies the calculation of angles, especially in calculus.

When using trigonometric functions, the angle's measurement unit—degrees or radians—impacts the outputs and graphs of these functions. For instance, the standard output range for the inverse cosine function is usually from 0 to \(\pi\) radians, illustrating a half-circle.

Radian

A radian is a unit of angular measure used in many areas of mathematics. It is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
To better understand radians:

• One complete revolution around a circle corresponds to an arc length equal to the circle’s circumference, which leads to a full circle measuring \(2\pi\) radians.
• Since \(2\pi\) radians is equal to 360 degrees, we have the conversion: 1 radian equals approximately 57.2958 degrees.
• Therefore, \(\pi\) radians is equal to 180 degrees.

Radians arise naturally in mathematics and are the standard unit when working with the trigonometric functions in calculus because they simplify many equations and calculations.
When calculating the inverse cosine of a value, such as \(-0.75\), using a calculator in radian mode is advised as it aligns with the mathematical properties of trigonometric functions used in higher-level applications and solutions.