Understanding the Values of sin 15° and cos 75°: Simplified Trigonometric Identities

In trigonometry, understanding the values of specific angles like sin 15° and cos 75° is crucial for solving more complex problems and proofs. This article aims to demystify the values of these trigonometric functions with the aid of elementary identities from trigonometry. We will also discuss the relationship between sin 15° and cos 75°, and provide a step-by-step guide to simplifying and finding the values of these trigonometric expressions.

Trigonometric Identities and Simplification Techniques

Trigonometric identities are fundamental formulas that help in the simplification and evaluation of trigonometric functions. Two of the most useful identities are:

(sin(A - B) sin A cos B - cos A sin B)

(cos(A - B) cos A cos B sin A sin B)

These identities allow us to break down complicated angles into simpler ones, making the evaluation of trigonometric functions more tractable.

Calculating sin 15°

To find the value of (sin 15^circ), we will use the identity for the difference of two angles:

(sin 15^circ sin(45^circ - 30^circ))

Applying the identity:

(sin(45^circ - 30^circ) sin 45^circ cos 30^circ - cos 45^circ sin 30^circ)

Using the known values of trigonometric functions for 45° and 30°:

(sin 45^circ frac{1}{sqrt{2}}, cos 30^circ frac{sqrt{3}}{2}, cos 45^circ frac{1}{sqrt{2}}, sin 30^circ frac{1}{2})

(sin 15^circ frac{1}{sqrt{2}} cdot frac{sqrt{3}}{2} - frac{1}{sqrt{2}} cdot frac{1}{2})

(sin 15^circ frac{sqrt{3}}{2sqrt{2}} - frac{1}{2sqrt{2}} frac{sqrt{3} - 1}{2sqrt{2}})

(sin 15^circ frac{sqrt{3} - 1}{2sqrt{2}} frac{sqrt{6} - sqrt{2}}{4})

Calculating cos 75°

To find the value of (cos 75^circ), we can use the identity for the difference of two angles as well:

(cos 75^circ cos(90^circ - 15^circ))

Using the identity for the complement of an angle:

(cos(90^circ - x) sin x)

(cos 75^circ sin 15^circ frac{sqrt{6} - sqrt{2}}{4})

Relationship Between sin 15° and cos 75°

From the above calculations, we can observe that (sin 15^circ frac{sqrt{6} - sqrt{2}}{4}) and (cos 75^circ frac{sqrt{6} - sqrt{2}}{4}). This leads us to conclude that:

(sin 15^circ cos 75^circ)

This result is a direct consequence of the trigonometric identity that relates the sine and cosine of complementary angles. In fact, for any angle (theta), we have:

(sin(90^circ - theta) cos theta)

Calculation of sin 15° × cos 75°

To calculate the product (sin 15^circ times cos 75^circ), we can directly use the values we have found:

(sin 15^circ frac{sqrt{6} - sqrt{2}}{4})

(cos 75^circ sin 15^circ frac{sqrt{6} - sqrt{2}}{4})

(sin 15^circ times cos 75^circ left(frac{sqrt{6} - sqrt{2}}{4}right) times left(frac{sqrt{6} - sqrt{2}}{4}right))

( frac{(sqrt{6} - sqrt{2})^2}{16})

( frac{6 - 2sqrt{12} 2}{16})

( frac{8 - 4sqrt{3}}{16})

( frac{1 - sqrt{3}}{4})

( 0.05856) (approximately)

Conclusion

Understanding and applying trigonometric identities is a powerful tool in mathematics, particularly in solving trigonometric problems. The values of (sin 15^circ) and (cos 75^circ), and their relationship, are derived using elementary identities from trigonometry. This article provides a clear and concise guide to simplifying and calculating these values, along with the product of these trigonometric expressions.

For further exploration, you can explore more trigonometric identities and their applications in geometry, calculus, and physics. Understanding these concepts will greatly enhance your mathematical skills and problem-solving capabilities.