Problem 30 - Algebra and Trigonometry

Question

Use a calculator to find the value of each expression rounded to two decimal places. $$ \tan ^{-1}(-\sqrt{5061}) $$

Step-by-Step Solution

Answer

The value of this operation, rounded to two decimal places, should be -89.57 degrees or in radian -1.57 when a calculator is used.

Step 1: Evaluate the Square Root

First, calculate the square root of 5061 which gives us approximately 71.13.

Step 2: Apply the Negative Sign

Next, apply the negative sign to our result from the previous step. This means we are now working with -71.13.

Step 3: Compute the Inverse Tangent

Now we apply the inverse tangent function. On the calculator, this could be written as tan^-1(-71.13).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. It is essential for understanding various principles of geometry and is widely applied in science and engineering. The basic functions in trigonometry are sine (sin), cosine (cos), and tangent (tan). These functions are ratios of sides of a right-angled triangle and are used to represent angles.

The tangent function, specifically, represents the ratio of the opposite side to the adjacent side of an angle in a right-angled triangle. Understanding the tangent function is crucial when solving problems related to angles and sides of triangles, making it a foundation for anyone aiming to master trigonometry.

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc-functions, are the inverse operations of the basic trigonometric functions. They are used to determine the angles when the value of the trigonometric function is known. The inverse tangent function, represented as \( tan^{-1} \) or \( \text{arctan} \), gives the angle for which the tangent is the input value.

Understanding the Inverse Tangent Function

When dealing with the inverse tangent, or \( tan^{-1} \) function, you're essentially asking 'what angle gives me this tangent value?' For example, \( tan^{-1}(1) \) would yield \( \frac{\pi}{4} \) radians or 45 degrees, since the tangent of 45 degrees is 1. When using the inverse tangent function with negative inputs, like in the exercise with \( tan^{-1}(-\sqrt{5061}) \), the resulting angle typically lies in the fourth quadrant of the trigonometric circle, where tangent values are negative.

Calculator Use in Trigonometry

Calculators are indispensable tools in trigonometry, especially when dealing with complicated numbers or inverse trigonometric functions. To work with these functions, scientific calculators often have a designated button, usually labeled 'tan⁻¹' or 'arctan'.

• When computing the inverse tangent, one must ensure the calculator is set to the correct mode (degrees or radians) depending on the context of the problem.
• Modern calculators can handle complex expressions straight away, but it's always good practice to break down the problem as shown in the step by step solution of the exercise.
• Be mindful of the calculator’s limitations and the precision of your model when interpreting the results.

Rounded Values in Mathematics

Rounded values are approximations of numerical values to a specified number of digits, often to simplify computations and to present the results in a more understandable way. In mathematics, it is crucial to know when and how to round numbers without significantly altering the accuracy of your result.

Rounding is especially important when dealing with inverse trigonometric functions, where exact values are difficult to find due to the irrational numbers often involved. When a problem asks for a solution 'rounded to two decimal places' it directs us to keep two digits after the decimal point, which strikes a balance between precision and readability.

Significance of Rounding

While rounding makes numbers more manageable, it also introduces a small error, so it's necessary to understand the implications and not round numbers until the final answer if possible, to maintain the integrity of the calculation.