Problem 30 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \cos ^{-1} 0.26 $$

Step-by-Step Solution

Answer

The value of \(\cos^{-1}(0.26)\) rounded to two decimal places is approximately 74.47 degrees.

Step 1: Understanding Inverse Cosine

Inverse cosine function, denoted as \(\cos^{-1}\) or \(\arccos\), is used to determine the angle whose cosine is a given number. Here, the given number is 0.26 and we need to find the angle 'a', such that \(\cos(a) = 0.26\).

Step 2: Using the Calculator

Enter '0.26' into your calculator, then press the \(\cos^{-1}\) or \(\arccos\) button, depending on your calculator.

Step 3: Rounding the Result

The result given by the calculator needs to be rounded to two decimal places. Ensure you use the correct rounding rules: if the third decimal is equal to or more than 5, round up; if it’s less than 5, just remove it.

Key Concepts

Inverse Cosine

The inverse cosine, denoted as either \(\cos^{-1}\) or \(\arccos\), is a special function in trigonometry. It helps find the angle when we already know the cosine of that angle. Essentially, if you know \(\cos(a)\), you can use \(\cos^{-1}\) to discover the measure of angle \(a\).
For example, if \(\cos(a) = 0.26\), the inverse cosine of 0.26 provides the angle \(a\). This is extremely useful in many fields like physics and engineering, where determining angles from their cosine values aids in solving various problems. Be aware that inverse cosine only outputs angles in the range from \(0\) to \(\pi\) radians (or \(0\) to \(180\) degrees). This is because cosine values repeat beyond these limits, and we need a unique output. Therefore, whenever you encounter tasks involving inverse cosine, keep these constraints in mind to achieve accurate results.
To use the inverse cosine on a calculator, input the cosine value and then press the \(\cos^{-1}\) button. It's a straightforward process, but make sure your calculator is set to the correct mode — degrees or radians — depending on the context of your work.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and science. They express the relationships between the angles and sides of triangles, especially right-angled ones. The primary trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).
Cosine, which is part of our focus here, relates the adjacent side to the hypotenuse in a right triangle. The formula is simple: \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\). Because trigonometric functions are periodic, they repeat values at specific intervals, which is why inverse functions like \(\cos^{-1}\) are necessary to find original angles for given cosine values.
Other trigonometric functions include secant (\(\sec\)), cosecant (\(\csc\)), and cotangent (\(\cot\)). These are less commonly used but are equally important in advanced mathematical problems. Understanding and manipulating these functions are key skills for solving problems in trigonometry.

Rounding Numbers

Rounding numbers is a simple yet important skill, especially in mathematics where precision is crucial. When we round a number, we are essentially adjusting it to a desired number of decimal places.
For example, if rounding to two decimal places, look at the third decimal place. If it's 5 or more, increase the second decimal by one and drop the rest. If it's less than 5, just remove the additional decimal digits.

• Example 1: The number 1.234 rounded to two decimal places becomes 1.23, since 4 is less than 5.
• Example 2: The number 1.236 becomes 1.24, as 6 is more than 5.

It's crucial to apply the correct rounding method to ensure solutions are both precise and accurate. Perfecting this technique is incredibly useful across all math ventures, as well as in real-world applications, like financial calculations.