Rewrite Tan 15 Degrees Using Sum Or Difference Formula

Trigonometric identities, especially sum and difference formulas, are fundamental tools in mathematics, physics, and engineering. These identities allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. In this comprehensive guide, we will delve into the sum and difference formulas for tangent, illustrating their application with a specific example: rewriting ${\tan 15^{\circ}}$ using these formulas. This exploration will not only solidify your understanding of these critical identities but also enhance your problem-solving skills in trigonometry. Mastering these concepts is crucial for anyone looking to excel in mathematics and related fields, enabling the simplification and solution of complex trigonometric problems.

The Significance of Sum and Difference Formulas

Sum and difference formulas are essential tools in trigonometry because they allow us to express trigonometric functions of compound angles (angles that are sums or differences of other angles) in terms of trigonometric functions of the individual angles. This capability is particularly useful when dealing with angles that are not standard values (such as 30°, 45°, 60°, and 90°) but can be expressed as sums or differences of these standard angles. For instance, 15° can be written as 45° - 30°, and 75° can be expressed as 45° + 30°. By using sum and difference formulas, we can find the exact trigonometric values of these non-standard angles, which is invaluable in various applications.

These formulas are not just theoretical constructs; they have practical applications in numerous fields. In physics, they are used to analyze wave phenomena, such as the interference and diffraction of light and sound. In engineering, they are crucial for designing and analyzing systems involving oscillations and vibrations. Moreover, in mathematics itself, these formulas are fundamental for solving trigonometric equations, simplifying expressions, and proving other trigonometric identities. Understanding and applying these formulas effectively is therefore a cornerstone of mathematical and scientific proficiency. The ability to break down complex angles into simpler components and apply the appropriate sum or difference formula is a powerful technique for simplifying calculations and gaining deeper insights into trigonometric relationships.

Sum and Difference Formulas for Tangent

The tangent function, being the ratio of sine to cosine, has its own set of sum and difference formulas derived from the sine and cosine formulas. These formulas are:

• Sum Formula:

  ${ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} }$

• Difference Formula:

  ${ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} }$

These formulas allow us to express the tangent of the sum or difference of two angles in terms of the tangents of the individual angles. The sum formula involves adding the tangents of the individual angles in the numerator and subtracting their product from 1 in the denominator. Conversely, the difference formula involves subtracting the tangents of the individual angles in the numerator and adding their product to 1 in the denominator. These formulas are crucial for simplifying trigonometric expressions and solving equations, particularly when dealing with non-standard angles. Understanding the derivation and application of these formulas is essential for mastering trigonometry and its applications in various fields.

Rewriting ${\tan 15^{\circ}}$ Using the Difference Formula

To rewrite ${\tan 15^{\circ}}$ using a sum or difference formula, we first need to express 15° as a sum or difference of angles for which we know the exact tangent values. A common choice is to express 15° as the difference between 60° and 45°, both of which have well-known tangent values. Specifically, we have:

${ 15^{\circ} = 60^{\circ} - 45^{\circ} }$

Now, we can apply the difference formula for tangent:

${ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} }$

Substituting ${A = 60^{\circ}}$ and ${B = 45^{\circ}}$ into the formula, we get:

${ \tan 15^{\circ} = \tan(60^{\circ} - 45^{\circ}) = \frac{\tan 60^{\circ} - \tan 45^{\circ}}{1 + \tan 60^{\circ} \tan 45^{\circ}} }$

This expression represents ${\tan 15^{\circ}}$ in terms of the tangents of 60° and 45°, which are standard angles with known values. This step is crucial because it allows us to replace the tangent of a non-standard angle (15°) with an expression involving tangents of standard angles, which we can easily evaluate. The application of the difference formula here demonstrates its power in simplifying trigonometric calculations and solving problems involving angles that are not multiples of 30° or 45°. By expressing the target angle as a difference of standard angles, we can leverage the known trigonometric values of these standard angles to find the trigonometric value of the original angle.

Evaluating the Expression

Now that we have expressed ${\tan 15^{\circ}}$ as:

${ \tan 15^{\circ} = \frac{\tan 60^{\circ} - \tan 45^{\circ}}{1 + \tan 60^{\circ} \tan 45^{\circ}} }$

we can substitute the known values of ${\tan 60^{\circ}}$ and ${\tan 45^{\circ}}$. We know that:

${ \tan 60^{\circ} = \sqrt{3} }$

and

${ \tan 45^{\circ} = 1 }$

Plugging these values into the expression, we get:

${ \tan 15^{\circ} = \frac{\sqrt{3} - 1}{1 + \sqrt{3} imes 1} = \frac{\sqrt{3} - 1}{1 + \sqrt{3}} }$

This fraction can be further simplified by rationalizing the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which is ${1 - \sqrt{3}}$:

${ \tan 15^{\circ} = \frac{(\sqrt{3} - 1)(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} }$

Expanding the numerator and the denominator, we have:

${ \begin{aligned} \tan 15^{\circ} &= \frac{\sqrt{3} - 3 - 1 + \sqrt{3}}{1 - 3} \\ &= \frac{2\sqrt{3} - 4}{-2} \\ &= 2 - \sqrt{3} \end{aligned} }$

Thus, ${\tan 15^{\circ} = 2 - \sqrt{3}}$. This final result provides the exact value of the tangent of 15 degrees, demonstrating the utility of the sum and difference formulas in finding trigonometric values for non-standard angles. The process of rationalizing the denominator ensures that the final answer is in a simplified form, which is often preferred in mathematical expressions.

Conclusion

In conclusion, we have successfully rewritten ${\tan 15^{\circ}}$ using the difference formula for tangent and evaluated the expression to find its exact value. This process highlights the power and utility of sum and difference formulas in trigonometry. By expressing angles as sums or differences of standard angles, we can leverage our knowledge of trigonometric values for these standard angles to find values for a broader range of angles. This technique is not only useful for solving mathematical problems but also has significant applications in various scientific and engineering disciplines. Mastering these formulas and their applications is crucial for anyone seeking a deeper understanding of trigonometry and its role in the world around us.

Key Takeaways

• Sum and difference formulas are essential tools for simplifying trigonometric expressions.

• ${\tan 15^{\circ}}$ can be expressed as ${\tan(60^{\circ} - 45^{\circ})}$ and evaluated using the difference formula for tangent.

• The exact value of ${\tan 15^{\circ}}$ is ${2 - \sqrt{3}}$.

• Rationalizing the denominator is an important step in simplifying expressions involving radicals.

Further Exploration

To further enhance your understanding, consider exploring other applications of sum and difference formulas, such as simplifying complex trigonometric equations or proving other trigonometric identities. Additionally, you can investigate the applications of these formulas in fields like physics and engineering, where they are used to analyze wave phenomena and oscillatory systems. Practice applying these formulas to a variety of problems to solidify your skills and deepen your understanding of their power and versatility.