Problem 17 - Precalculus Mathematics for Calculus

Question

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\tan ^{-1} 10$$

Step-by-Step Solution

Answer

The approximate value of \( \tan^{-1}(10) \) is 1.47113 radians.

Step 1: Understand the Inverse Tangent Function

The inverse tangent function, written as \( \tan^{-1} \) or \( \arctan \), gives the angle whose tangent is a given number. Here, we want to find the angle \( \theta \) such that \( \tan(\theta) = 10 \).

Step 2: Set Up Your Calculator

Ensure your calculator is in radians mode as angles are commonly expressed in radians for trigonometric functions. This is crucial to get the correct result.

Step 3: Use the Calculator to Find the Value

Enter \( \tan^{-1}(10) \) into your calculator. The calculator computes the angle whose tangent is 10.

Step 4: Record the Result

The calculator provides a numerical result. Make sure to round this result to five decimal places. According to the calculator, \( \tan^{-1}(10) \approx 1.47113 \) radians.

Key Concepts

Trigonometric Identities

Trigonometric identities help simplify and solve various trigonometric expressions and equations. They are rules that relate the angles and sides of triangles, and are foundational in trigonometry.

• Basic Identities: There are six basic trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).
• Reciprocal Identities: These identities express each function in terms of others, like \(\csc\theta = \frac{1}{\sin\theta}\).
• Pythagorean Identities: These are based on the Pythagorean theorem, such as \(\sin^2\theta + \cos^2\theta = 1\).
• Inverse Functions: Inverse trigonometric functions, like \(\tan^{-1}\), retrieve the angle corresponding to a given trigonometric value. For instance, if \(\tan(\theta) = 10\), then \(\theta = \tan^{-1}(10)\).

Understanding these identities helps in evaluating expressions and solving equations, as they are used to change complex trigonometric expressions into simpler forms.

Radians and Degrees

Angles can be measured in radians or degrees, with both being standard units for expressing angles. Understanding both units is essential in trigonometry.

• Degrees: A more familiar unit, a circle consists of 360 degrees. It’s often used in everyday contexts, like measuring angles in geometry.
• Radians: Radians, a standard unit in mathematical contexts, arise from the relationship between the radius of a circle and the arc length. A complete circle is \(2\pi\) radians, equating a full rotation to \(360^\circ\).

To convert between degrees and radians:

• From degrees to radians: Multiply by \(\frac{\pi}{180}\).
• From radians to degrees: Multiply by \(\frac{180}{\pi}\).

When solving trigonometric problems, especially when using calculators, make sure to set your calculator to the appropriate mode (radians or degrees) to get accurate results.

Rounding Numbers

Rounding numbers is a critical skill for ensuring precision without an overload of unnecessary detail. In mathematics, it's often important to specify results to a required precision.

• Basic Rounding: Rounding adjusts a number to a specified degree of accuracy. For example, 1.471128 rounded to five decimal places becomes 1.47113.
• Rules for Rounding: Check the digit immediately following the required precision:
  • If it's 5 or greater, increase the last retained digit by 1.
  • If it's less than 5, simply remove subsequent digits.

Rounding is vital in presenting results from calculations. It keeps computations concise and avoids the clutter that excessive detail can cause.
It is essential when reporting results, ensuring consistency, and maintaining clear communication in mathematics and its applications.