Problem 14 - Algebra 2

Question

Use a calculator and inverse functions to find the radian measures of the angles. angles whose tangent is 5

Step-by-Step Solution

Answer

The radian measure of the angle whose tangent is 5 is the numeric value obtained from the calculator (approximated value: 1.373 radians)

Step 1: Identify the Known Value

From the exercise, it is known that the tangent of the angle equals to 5. Therefore, the value for which the inverse tangent is to be found is 5.

Step 2: Find the Inverse Tangent

As the task provides the tangent of the angle and you need to find the angle, use a calculator to find the inverse (arctangent) of 5. It's important to ensure that the calculator is in radian mode. Enter '5' and then press the inverse tangent (often represented as 'tan^-1') button on your calculator.

Step 3: Write Down the Result

The result from the calculator is the radian measure of the desired angle. Make sure to jot it down.

Key Concepts

Understanding Radian Measure

Radians are a unit of angular measurement used mainly in trigonometry, specifically involving circular and periodic motion. Unlike degrees, which divide a circle into 360 segments, radians divide the circle into just over six segments. One full circle is equal to \(2\pi\) radians.

• Why radians? They provide a direct link between linear and angular motion, which is essential for calculus and advanced mathematics.
• Converting between degrees and radians: to convert degrees to radians, use the formula \(\theta \, \text{radians} = \theta \, \text{degrees} \times \frac{\pi}{180}\).
• For example, 90° is \(\frac{\pi}{2}\) in radians.

Understanding when and how to use radians can particularly enhance calculations in calculus, particularly when dealing with periodic functions like sine, cosine, and tangent.

Explaining the Tangent Function

The tangent function is one of the basic functions in trigonometry. It relates an angle of a right triangle to the ratio of the opposite side over the adjacent side.

• In mathematical terms, if \(\theta\) is an angle in a right triangle, then \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
• Tangent values are periodic, repeating every \(\pi\) radians.
• Unlike sine and cosine, tangent can be greater than 1 or less than -1, theoretically spanning from negative to positive infinity.

The tangent function is particularly useful in problems involving angles where ratios rather than specific side lengths are known, such as the one in our exercise.

Using a Calculator for Trigonometric Functions

Calculators are extremely helpful when working with trigonometric functions, especially for quickly finding values like the arctangent. Using a calculator with the correct mode set is crucial.

• Ensure the calculator is in radian mode. Often indicated by an icon or a mode setting, radians make a difference in calculations.
• To find the arctangent: input the number whose tangent was initially computed. So, input "5" as described in the exercise.
• Press the button often labeled as \(\tan^{-1}\) which may appear on calculators as "inv tan" or "atan". The result will be an angle in radians.

Using the calculator accurately helps confirm that computations are correct, such as ensuring there's no error when switching between modes, especially when dealing with inverse trigonometric functions.

Understanding Arctangent

Arctangent is an inverse trigonometric function. It allows us to find the angle which corresponds to a tangent value.

• If \(\tan(\theta) = x\), then \(\arctan(x) = \theta\).
• The range of the arctangent function: the resulting angle from the arctangent function will lie between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians.
• In practical terms, when given a tangent, using arctangent helps to backtrack to find the actual angle.

In exercises like the one presented, finding arctangent using a calculator simplifies the process of uncovering the angle from its trigonometric ratio involvements.