Problem 43 - Algebra 2

Question

Use the sum and difference formulas for sine and cosine to derive formulas for \(\tan (\alpha+\beta)\) and \(\tan (\alpha-\beta)\)

Step-by-Step Solution

Answer

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

Step 1: Recall the Sine and Cosine Sum Formulas

The sum formulas for sine and cosine are given by:- \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \)- \( \cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta \)

Step 2: Express Tangent Using Sine and Cosine

Tangent is expressed as the ratio of sine to cosine. Therefore:\[ \tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} \]

Step 3: Apply the Sine and Cosine Formulas to Tangent

Substitute the expressions for \( \sin(\alpha + \beta) \) and \( \cos(\alpha + \beta) \) into the tangent formula:\[ \tan(\alpha + \beta) = \frac{\sin\alpha \cos\beta + \cos\alpha \sin\beta}{\cos\alpha \cos\beta - \sin\alpha \sin\beta} \]

Step 4: Simplify the Tangent Expression

Now, simplify the expression assuming \( \cos\alpha \cos\beta - \sin\alpha \sin\beta eq 0 \). This becomes:\[ \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta} \] by dividing the numerator and denominator by \( \cos\alpha \cos\beta \).

Step 5: Recall the Sine and Cosine Difference Formulas

The difference formulas for sine and cosine are given by:- \( \sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta \)- \( \cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta \)

Step 6: Apply the Sine and Cosine Formulas to Tangent Difference

Substitute the expressions for \( \sin(\alpha - \beta) \) and \( \cos(\alpha - \beta) \) into the tangent formula:\[ \tan(\alpha - \beta) = \frac{\sin\alpha \cos\beta - \cos\alpha \sin\beta}{\cos\alpha \cos\beta + \sin\alpha \sin\beta} \]

Step 7: Simplify the Tangent Difference Expression

Now, simplify the expression assuming \( \cos\alpha \cos\beta + \sin\alpha \sin\beta eq 0 \). This becomes:\[ \tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha \tan\beta} \] by dividing the numerator and denominator by \( \cos\alpha \cos\beta \).

Key Concepts

Sum and Difference Formulas

The sum and difference formulas are the backbone of many trigonometric identities, allowing us to express trigonometric functions of sums or differences of angles in terms of functions of single angles. For sine and cosine, these formulas are crucial for simplifying complex trigonometric expressions.

When dealing with the sum of two angles, \( \), the formulas are:

• \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \)
• \( \cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta \)

Similarly, for the difference of two angles, \( \alpha - \beta \):

• \( \sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta \)
• \( \cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta \)

These formulas are derived from the unit circle and can be used to find the sine and cosine of any angle combination. They are extensively used in transforming expressions and solving trigonometric equations.

Tangent Function

The tangent function is another fundamental trigonometric function representing the ratio of sine to cosine:
\( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

To find the tangent of sum or difference of two angles, \( \alpha + \beta \) or \( \alpha - \beta \), we employ both the sum and difference formulas for sine and cosine. This involves substituting the results of the sine and cosine sum and difference into the tangent function:

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

By plugging in the respective formulas for sine and cosine, and further simplifying by dividing both numerator and denominator by appropriate terms, one derives the efficient tangent sum and difference identities:

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

These identities simplify calculations involving tangent functions considerably, especially when angles are not standard.

Sine and Cosine Formulas

Sine and cosine formulas are among the most useful in trigonometry for breaking down complex angle expressions into more manageable pieces.

When using these formulas, it is crucial to remember how sine and cosine change based on angle sums and differences. They allow transformations between different angles to be calculated easily and help in proving identities.

For example, the formulas for the sums and differences:

• Sum:
  • \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \)
  • \( \cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta \)
• Difference:
  • \( \sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta \)
  • \( \cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta \)

These allow for precise calculations and transformations based on simple trigonometric functions. Understanding these concepts fully will greatly enhance one's ability to solve trigonometric problems effectively.