Problem 36 - Contemporary Precalculus

Question

(a) Show that \(\tan (t+2 \pi)=\tan t\) for every \(t\) in the domain of tan \(t .\) [ Hint: Use the definition of tangent and some identities proved in the text. \(]\) (b) Verify that it appears true that \(\tan (x+\pi)=\tan x\) for every \(t\) in the domain by using your calculator's table feature to make a table of values for \(y_{1}=\tan (x+\pi)\) and \(y_{2}=\tan x\)

Step-by-Step Solution

Answer

Question: Prove that \(\tan(t+2\pi)=\tan(t)\) for every value of \(t\) and verify it using a calculator. Answer: We have proved that \(\tan(t+2\pi)=\tan(t)\) using the sum of angles formula for tangent and showed that the identity is true for all values of \(t\). To verify it using a calculator, compare the values of \(\tan(x+\pi)\) and \(\tan(x)\) for different \(x\) values using a table feature in the calculator. If the corresponding values are the same or very close in value (due to rounding errors), then the identity appears to be true.

Step 1: Sum of angles formula for tangent

Recall the sum of angles formula for tangent which states that: \(\tan(a+b) = \frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}\)

Step 2: Use the formula for the given problem

Use the formula to find the tangent of the sum of the angles: \(\tan(t+2\pi) = \frac{\tan(t)+\tan(2\pi)}{1-\tan(t)\tan(2\pi)}\)

Step 3: Identify the tangent of 2π

Use the identity of the tangent of the angle \(2\pi\), which is: \(\tan(2\pi) = 0\) This is because \(\sin(2\pi) = 0\) and \(\cos(2\pi) = 1\), so \(\tan(2\pi) = \frac{\sin(2\pi)}{\cos(2\pi)} = 0\).

Step 4: Substitute the value of \(\tan(2\pi)\) in the equation

Now, we can substitute the value of \(\tan(2\pi)\) into the equation from step 2: \(\tan(t+2\pi) = \frac{\tan(t)+0}{1-\tan(t)(0)}\)

Step 5: Simplify the expression

Simplify the expression: \(\tan(t+2\pi) = \frac{\tan(t)}{1} = \tan(t)\) So we have shown that \(\tan(t+2\pi)=\tan(t)\) for every value of \(t\). #Phase 2: Verifying the identity using a calculator#

Step 1: Set up calculator table

Set up a table with the following values in a calculator: - Column 1 (\(x\)): Choose some values for \(x\). - Column 2 (\(y_{1}\)): Compute the values of \(\tan(x+\pi)\) for each value of \(x\). - Column 3 (\(y_{2}\)): Compute the values of \(\tan(x)\) for each value of \(x\).

Step 2: Compare values of \(y_{1}\) and \(y_{2}\)

Compare the corresponding values of \(y_{1}\) and \(y_{2}\). If they are the same or very close in value (due to rounding errors), then we can say that the identity appears to be true. By following these steps, you should be able to verify that \(\tan(x+\pi)=\tan(x)\) for every value of \(t\) in the domain using a calculator's table feature.

Key Concepts

Trigonometric Identities

Trigonometric identities are the backbone of solving many problems in trigonometry. They act as tools that allow us to simplify complex expressions or prove certain relationships between trigonometric functions. For instance, one of the basic trigonometric identities is the Pythagorean identity, which states that \(\sin^2(x) + \cos^2(x) = 1\), relating the sine and cosine of the same angle.

Another important set of identities are the quotient identities which provide a way to transform the tangent function in terms of sine and cosine. The tangent of an angle is defined as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). Knowing these identities allows us to approach tangent-related problems more strategically, as we can convert tangents into simpler terms when necessary.

Angle Sum Formula

The angle sum formula for tangent is particularly useful when dealing with expressions that involve the sum (or difference) of two angles. The formula is given by \(\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\). This relationship helps us to find the tangent of a composite angle based on the tangents of its individual parts.

Using the angle sum formula is essential when simplifying expressions like \(\tan(t+2\pi)\) or proving periodicity properties. It allows us to acknowledge that adding a full rotation (\(2\pi\) radians) to an angle does not change the tangent's value, hence demonstrating the periodic nature of the tangent function.

Tangent Periodicity

Periodicity is a fundamental characteristic of the trigonometric functions. The tangent function is periodic with a period of \(\pi\), meaning that it repeats its values every \(\pi\) radians. This periodicity is a crucial concept when working with trigonometric functions since it helps predict the behavior of these functions over different intervals.

The identity \(\tan(t+2\pi) = \tan(t)\) is a direct result of this periodic nature. Since \(2\pi\) is twice the tangent’s period, adding it to any angle value \(t\) does not alter the function's outcome. Understanding the periodicity of tangent can also aid in graphing the function and solving equations involving the tangent.

Calculator Table Feature

Modern calculators often come with a table feature, which can greatly assist in the study of functions, particularly in verifying trigonometric identities. By setting up a table with angles and their corresponding trigonometric function values, students can analyze and verify identities practically.

For verifying an identity such as \(\tan(x + \pi) = \tan(x)\), a side-by-side comparison of the function values at different angles may be made. Students should input a range of \(x\) values in the calculator and then compute both \(\tan(x + \pi)\) and \(\tan(x)\) for each one. If the corresponding outputs match or are very close (considering round-off errors), the identity's validity is supported in a tangible manner.