Problem 28 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arctan 25 $$

Step-by-Step Solution

Answer

The answer is \(\approx 88.38\) degrees

Step 1: Compute the arctan function

Use a scientific calculator and find the arctan function, input the value 25.

Step 2: Use the result and round to two decimal places

After getting the result, make sure to round it to the second decimal place according to the classical rounding rules.

Key Concepts

Scientific Calculator Usage

When working with trigonometric problems such as evaluating the arctan of a number, a scientific calculator is an invaluable tool. To use one effectively, first ensure that the calculator is in the correct mode (degrees or radians) for your problem. Next, locate the arctan (often labeled as 'tan^-1') function button.

Enter the number you're evaluating, in this case, 25, and press the arctan function button. The calculator will display the angle whose tangent is 25. This process simplifies solving trigonometric problems without relying on trigonometry tables or manual calculations. For a step-by-step approach, be methodical: input the value accurately, use the correct function, and double-check your process for potential errors to ensure a precise outcome.

Trigonometric Functions

Trigonometric functions are foundational tools in mathematics, particularly useful in various applications such as physics, engineering, and even navigation. These functions relate the angles of a right triangle to the lengths of its sides.

The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), alongside their reciprocal functions: cosecant (csc), secant (sec), and cotangent (cot). The function we are focusing on here, arctan, is related to the tangent function, which is the ratio of the opposite side over the adjacent side of a right triangle. Understanding each of these functions and their relationships is crucial for solving trigonometric equations and modeling periodic phenomena.

Rounding Decimals

Rounding decimals is a significant numerical skill, essential for presenting data in a more understandable form, especially when precision is less critical. To round a number to two decimal places, you look at the third decimal place.

If this third decimal place is 5 or more, you increase the second decimal by one (this is known as rounding up). If it is less than 5, you leave the second decimal place as it is (rounding down). For example, if your calculator shows an arctan value of 1.6789, you would round this to 1.68 as the third decimal place is less than 5. This mastery of rounding rules ensures accurate representation of numerical answers, which is particularly important in scientific and mathematical reporting.

Inverse Trigonometric Functions

Inverse trigonometric functions, sometimes called arcfunctions, allow us to find the angle when the value of the trigonometric function is known.

The arctan function, specifically, gives us the angle whose tangent value equals the input number. It's the inverse of the tangent function. When we calculate the arctan of 25, we are essentially asking for the angle in a right triangle where the opposite side is 25 times longer than the adjacent side. These inverse functions open up possibilities for solving triangles when certain side lengths are known and are widely used in calculus and analytical geometry to find angles with given trigonometric values.