Problem 40 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \tan ^{-1}(-\sqrt{2165}) $$

Step-by-Step Solution

Answer

The result of \(\tan^{-1}(-\sqrt{2165})\), when rounded to two decimal places, will depend on the specific value obtained using the scientific calculator.

Step 1: Understanding the Expression

The given exercise is \(\tan^{-1}(-\sqrt{2165})\). Here, \(\tan^{-1}\) denotes the inverse tangent function, also known as arctangent. This function gives the angle whose tangent is the number inside the brackets. In this case, the number inside is -\(\sqrt{2165}\). So the task is to find the angle, whose tangent is -\(\sqrt{2165}\).

Step 2: Find the Square Root and Calculate the Tangent Inverse

Firstly, calculate the square root of 2165. Then, get the negative result of square root. Lastly, calculate the inverse tangent of the result. You can achieve these using a scientific calculator.

Step 3: Rounding the Result

The final step is to round the resulting angle to the nearest two decimal places. Use the rounding rules for this, where if the third decimal place is greater than or equal to 5, the second decimal place is rounded up.

Key Concepts

Arctangent

The arctangent, often represented as \( \tan^{-1} \), is an important concept in trigonometry. It is the inverse function of the tangent function. This means it is used to determine the angle that corresponds to a given tangent value. For instance, if \( \tan(\theta) = x \), then \( \tan^{-1}(x) = \theta \).

• The range of arctangent is from \(-\pi/2\) to \(\pi/2\) radians.
• Arctangent helps solve problems requiring the finding of angles from the known tangent values.
• In many calculators, the function is labeled as "atan" or "tan^-1".

Arctangent is especially useful in cases where the angle cannot be easily derived from direct trigonometric relationships.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, since \(3 \times 3 = 9\). In mathematical notation, the square root of 2165 is \(\sqrt{2165}\).

• Finding the square root is fundamental in simplifying expressions.
• Negative values of a square root indicate an additional step; in this problem, \(-\sqrt{2165}\) represents the negative of the square root of 2165.
• Square roots can be approximated using calculators if they do not resolve to whole numbers.

Calculating accurately is crucial because subsequent steps, like finding the arctangent, depend on this value.

Tangent Function

The tangent function is one of the basic trigonometric functions, often symbolized as \( \tan \). It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.

• Its value increases as the angle approaches \(90^o\) and decreases towards \(0^o\).
• The function can take any real number and returns a periodic output based on its input angle.
• Zer0 angles result in a tangent of 0, but as it reaches \(\pm90^o\), it becomes undefined due to division by zero.

The tangent's behavior deeply informs the results from the arctangent, as seen in the given exercise where tangent values guide which results are valid.

Scientific Calculator

Scientific calculators are essential tools for carrying out complex calculations, often needed for solving expressions like \( \tan^{-1}(-\sqrt{2165}) \).

• These calculators can perform square roots, trigonometric functions, and their inverses.
• Entering values must be accurate - it typically involves inputting the square root function, the negative sign, and then applying the arctangent.
• They allow rounding features, letting users easily round results to desired decimal places.

Using a scientific calculator correctly ensures another layer of precision, critical for getting to the right answer efficiently.