Problem 56 - Precalculus Essentials

Question

Derive the identity for \(\tan (\alpha-\beta)\) using $$\tan (\alpha-\beta)=\tan [\alpha+(-\beta)]$$ After applying the formula for the tangent of the sum of two angles, use the fact that the tangent is an odd function.

Step-by-Step Solution

Answer

The identity to be derived is \( \tan (\alpha-\beta)=\frac{\tan\alpha -\tan\beta}{1+\tan\alpha \tan\beta}\)

Step 1: Use the formula for the sum of two angles

Begin by expressing \( \tan (\alpha - \beta) \) as \( \tan [\alpha+(-\beta)] \). Now, apply the formula for the tangent of the sum of two angles, which is given as \( \tan(\alpha+\beta)=\frac{\tan\alpha + \tan\beta}{1-\tan\alpha \tan\beta}\) . Therefore, \( \tan [\alpha+(-\beta)]=\frac{\tan\alpha - \tan\beta}{1+\tan\alpha \tan\beta}\).

Step 2: Apply the property that the tangent is an odd function

Given that the tangent is an odd function, we know that \( \tan(-x)=-\tan(x) \). Therefore, we correct equation from the step 1 into \( \tan (\alpha-\beta)=\frac{\tan\alpha + \tan(-\beta)}{1-\tan\alpha \tan(-\beta)} \). Now, substitute \( -\tan\beta \) for \( \tan(-\beta) \) to get \( \tan (\alpha-\beta)=\frac{\tan\alpha -\tan\beta}{1+\tan\alpha \tan\beta}\) .

Step 3: Conclude the identity

Thus, the desired formula for \( \tan(\alpha-\beta) \) has been derived, which is \( \tan (\alpha-\beta)=\frac{\tan\alpha -\tan\beta}{1+\tan\alpha \tan\beta}\)

Key Concepts

Tangent Function

The tangent function, commonly denoted as \( \tan \theta \), is one of the fundamental trigonometric functions. It is defined as the ratio of the sine to the cosine of an angle in a right-angled triangle.
This means for a given angle \( \theta \), the tangent is calculated as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function has several distinctive properties:

• Periodicity: The tangent function repeats its values in intervals of \( \pi \) since \( \tan(\theta + \pi) = \tan \theta \).
• Even-Odd Function: It is an odd function, meaning that \( \tan(-\theta) = -\tan(\theta) \), which has implications in many trigonometric identities and simplifies calculations.
• Undefined Points: At certain angles like \( 90^\circ \) or \( 270^\circ \), the tangent function becomes undefined since \( \cos \theta = 0 \).

Knowing these properties helps in deducing and understanding various trigonometric identities.

Angle Subtraction Formula

The angle subtraction formula for the tangent function is an important identity used in trigonometry. It helps calculate the tangent of the difference of two angles. This formula can be derived using the properties of trigonometric functions and the concept that the tangent function is odd.
The formula is as follows:

• \( \tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta} \)

To derive it:- Represent \( \tan(\alpha - \beta) \) as \( \tan(\alpha + (-\beta)) \).- Utilize the tangent addition identity: \( \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} \).- Replace \( \beta \) with \( -\beta \), which through the odd function property gives \( \tan(-\beta) = -\tan\beta \).- Substitute and simplify using \( -\tan\beta \).This gives you the subtraction formula, allowing for the evaluation of tangents when angles are subtracted.

Trigonometric Functions

Trigonometric functions are tools used to relate angles and sides of a triangle. These functions include sine, cosine, tangent, cotangent, secant, and cosecant. They are fundamental in various fields such as geometry, physics, and engineering.
Here's a brief overview:

• Sine (\( \sin \theta \)): Ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)): Ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): Ratio of the opposite side to the adjacent side.

Additional functions include:

• Cotangent (\( \cot \theta \)): Reciprocal of tangent \( \left( \frac{1}{\tan \theta} \right) \).
• Secant (\( \sec \theta \)): Reciprocal of cosine \( \left( \frac{1}{\cos \theta} \right) \).
• Cosecant (\( \csc \theta \)): Reciprocal of sine \( \left( \frac{1}{\sin \theta} \right) \).

These functions are vital in understanding waves, oscillations, and periodic phenomena due to their cyclical nature. Knowing how these functions work enables solving complex mathematical problems.