Problem 29 - Algebra and Trigonometry

Question

Use a calculator to find the value of each expression rounded to two decimal places. $$ \tan ^{-1}(-\sqrt{473}) $$

Step-by-Step Solution

Answer

After following step 1, your calculator should provide the value of \(\tan^{-1}(-\sqrt{473})\), rounded off to two decimal places.

Step 1: Calculate the inverse tangent

Use your scientific calculator to find the value of \(\tan^{-1}(-\sqrt{473})\). Please ensure to input the negative sign before entering the square root function. To compute the square root function, enter '473' and choose the sqrt function. You then select the 'tan-1' function and insert the value you just computed.

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

After following step 1, your calculator should provide the value of \(\tan^{-1}(-\sqrt{473})\), rounded off to two decimal places.

Key Concepts

Inverse Tangent Calculation

The inverse tangent function, denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is used to find an angle whose tangent is \( x \). It is the inverse operation of the tangent function in trigonometry. When performing an inverse tangent calculation, you are essentially asking: "What angle has this particular tangent value?"
In our exercise, we need to determine the angle with the tangent \( -\sqrt{473} \). The negative sign indicates that the angle is in the fourth or second quadrant if using degrees. Calculators can effortlessly handle such operations, providing an angle in either degrees or radians based on your settings. Always make sure to consider the context or the settings required for solving such mathematical problems.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and include the sine, cosine, and tangent functions, along with their inverses. These functions are based on the relationships between the angles and sides of right-angled triangles.

• The tangent function relates an angle in a right triangle to the ratio of the opposite side to the adjacent side.
• Its inverse, \( \tan^{-1} \), helps calculate the angle when the tangent is known.

Understanding these functions is crucial, as they have wide-ranging applications in physics, engineering, and computer science.
Since it can be hard to grasp, remember that trigonometry often deals with ratios and angles. Knowing how to switch between the function and its inverse is key to solving related problems.

Scientific Calculator Usage

A scientific calculator is an essential tool for solving advanced mathematical problems, like calculating inverse trigonometric functions. These calculators have functions for trigonometry, including \( \tan^{-1} \), which are typically accessed via a dedicated button or a secondary function key.
To use a scientific calculator for an exercise like \( \tan^{-1}(-\sqrt{473}) \):

• First, input '473' and then use the square root function.
• Be careful to include the negative sign at the start. Most calculators allow you to input the minus sign directly, but it is crucial to enter it before calculating the square root.
• Afterward, press the \( \tan^{-1} \) function key to find the inverse tangent.

It's important to familiarize yourself with your specific calculator model. Reading the manual can help you understand any unique features or key combinations. With practice, these calculations become faster and more intuitive.

Mathematical Expressions

Mathematical expressions involve numbers, variables, and operators (like addition or multiplication) to express calculations or relationships concisely. Understanding how to read and interpret these expressions is essential in performing calculations accurately.
In this exercise, the expression \( \tan^{-1}(-\sqrt{473}) \) involves multiple components:

• The square root, represented by \( \sqrt{473} \), is the operation you perform first.
• The negative sign, \(-\), affects the direction and quadrant of the resulting angle.
• Lastly, the inverse tangent function \( \tan^{-1} \) finds the angle corresponding to the tangent of \(-\sqrt{473}\).

This order of operations is crucial and should follow proper mathematical conventions to arrive at the correct result. Practice with various expressions helps in mastering complex calculations efficiently.