Problem 13 - Precalculus Mathematics for Calculus

Question

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\cos ^{-1}\left(-\frac{3}{7}\right)$$

Step-by-Step Solution

Answer

The approximate value is 1.98231 radians.

Step 1: Recognize the Problem

The problem is asking us to find the inverse cosine, also known as arc cosine \( \cos^{-1} \), of \( -\frac{3}{7} \). This means we need to determine the angle whose cosine is \( -\frac{3}{7} \).

Step 2: Understand the Domain and Range

The function \( \cos^{-1} \) is defined for values in the range \([-1, 1]\) and returns values in the range \([0, \pi]\) in radians, or \([0, 180]\) in degrees.

Step 3: Use a Calculator

Using a calculator, input \( \cos^{-1}\left(-\frac{3}{7}\right) \). Ensure the calculator is in radian mode if needed, as the answer should initially be considered in radians.

Step 4: Interpret the Calculator's Output

The calculator will give you an approximate value that is the angle in radians. Ensure the value is rounded to five decimal places.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics that relate an angle of a right triangle to the ratios of two side lengths. The primary trigonometric functions include sine ( \(\sin\) ), cosine ( \(\cos\) ), and tangent ( \(\tan\) ). These functions help describe the properties of various shapes and are widely used in fields like physics, engineering, and computer science.

• Sine: It represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine: This is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent: It gives the ratio of the length of the opposite side to the adjacent side.

Understanding these functions is crucial for solving geometrical problems and for analyzing periodic phenomena, such as waves. In our exercise, the inverse of the cosine function is used to find the angle when we know the cosine value. This operation helps us understand angles better when dealing with non-right triangles or complex periodic functions.

Radian Measure

Radian measure is a way of expressing angles. Instead of measuring angles in degrees, we measure them based on the radius of the circle. This is a more natural and mathematical way of thinking about angles.

• One radian is the angle created when the arc length is equal to the radius of the circle.
• The full circle, in radians, is \(2\pi\) , which is equivalent to 360 degrees.

A radian gives a clear and simple relationship between an angle and the arc it subtends. This makes mathematical calculations involving angles much easier. Radian measure is particularly convenient in trigonometry and calculus as it can simplify complex equations and functions. When using calculators for trigonometric functions, it is common and often necessary to set them to radian mode to ensure accuracy, especially in academic settings.

Arc Cosine

Arc cosine, denoted as \(\cos^{-1}\) , is the inverse function of the cosine. It computes the angle whose cosine is a given number. Remember, the domain for the \(\cos^{-1}\) function is \([-1, 1]\) , meaning it can only accept values within this range.

• The output of \(\cos^{-1}\) is an angle, usually expressed in radians, which falls between \(0\) and \(\pi\) .
• It is helpful for finding angles in situations like the exercise you encountered, where the cosine value ( \(-\frac{3}{7}\) ) is given.

Arc cosine is widely used in mathematics and scientific fields for solving equations where angles are unknown but can be calculated from known sine, cosine, or tangent values. Its understanding is crucial for anyone dealing with angles and trigonometric functions.