Problem 25 - Precalculus Essentials

Question

Use a calculator to find the value of each expression rounded to two decimal places. $$\cos ^{-1} \frac{\sqrt{5}}{7}$$

Step-by-Step Solution

Answer

After calculation, rounding the result to two decimal places gives the final answer.

Step 1: Understanding the problem

This problem requires finding the value of the inverse cosine (cos^{-1}) of a number, which in this case is \(\frac{\sqrt{5}}{7}\). The inverse cosine of a number is the angle whose cosine is that number.

Step 2: Evaluating the expression

Input.the given expression \(\cos^{-1}(\frac{\sqrt{5}}{7})\) in the calculator to get the result.

Step 3: Rounding the result

The value obtained from the calculator is then rounded to two decimal places as directed in the exercise.

Key Concepts

Cosine Function

The cosine function is one of the six fundamental trigonometric functions, and it plays a key role when dealing with angles and triangle measurements. In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. This makes it particularly useful in real-life applications such as physics and engineering. The cosine function is periodic and even, with a period of \(2\pi\), and it repeats every full circle around the unit circle.
Consider the expression \(\cos\theta = x\), where \(\theta\) is an angle, and \(x\) is the ratio. The inverse cosine function, or \(\cos^{-1}(x)\), allows us to find the angle \(\theta\) if we know the cosine value. This is crucial in solving problems like the one given in the exercise, where the inverse cosine is used to find the angle associated with a particular cosine value.
Understanding the cosine function gives you insight into how angles are measured and helps in interpreting many phenomena involving oscillations, waves, and circles.

Angle Measurement

Angles are measured in degrees or radians, which are two units of angular measurement. Degrees are more common in everyday situations, while radians are used primarily in calculus and higher mathematics. One full revolution around a circle measures \(360^\circ\) or \(2\pi\) radians.
When dealing with inverse trigonometric functions like the inverse cosine, the result is usually an angle. In this context, specifically for the inverse cosine function, the output is typically measured in radians. The range of \(\cos^{-1}(x)\) is from \(0\) to \(\pi\) radians, or from \(0^\circ\) to \(180^\circ\). This range is important because it helps us understand which quadrant the angle is in, ensuring that its cosine value falls within the possible range.
Grasping the concept of angle measurement can help solve many mathematical problems, including those involving navigation, astronomy, and architecture.

Decimal Approximation

Decimal approximation is essential in mathematics, especially when dealing with non-exact numbers. Many trigonometric calculations result in irrational numbers, which cannot be expressed exactly as a finite or repeating decimal.
In the context of the exercise, decimal approximation involves rounding the result of \(\cos^{-1}(\frac{\sqrt{5}}{7})\) to two decimal places. This means identifying the third decimal number and rounding based on its value: round up if it's 5 or more, and down if it's less than 5. This simplification makes numerical results more manageable while maintaining reasonable accuracy.
Approximating decimals is vital in practical applications, ensuring that calculations can be easily communicated and understood in everyday contexts, such as engineering, economics, and technology.