Introduction

In this article, we will delve into the value of sin?15° and explore multiple methods to calculate it accurately. This value is often used in trigonometric calculations and understanding its derivation can provide valuable insights into the field of trigonometry.

Trigonometric Methods for Finding sin 15°

The value of sin?15° can be derived using a variety of methods, all of which rely on basic trigonometric identities and formulas.

Using the Sine Subtraction Formula

The sine subtraction formula is a useful tool for finding the sine of an angle that can be expressed as the difference of two known angles. Specifically, we can express 15° as 45° - 30° and use the formula:

(sin(a - b) sin(a)cos(b) - cos(a)sin(b))

Let's substitute a 45° and b 30°. Using the known values of the trigonometric functions:

sin(45°) 2cos(30°) 2cos(45°) 2sin(30°) 12

Substituting these values into the formula, we get:

(sin(15°) left(frac{sqrt{2}}{2} cdot frac{sqrt{3}}{2}right) - left(frac{sqrt{2}}{2} cdot frac{1}{2}right))

Cancelling out the denominators, we find:

(sin(15°) frac{sqrt{6} - sqrt{2}}{4})

Alternative Method Using Triangle Geometry

Another method involves constructing a right triangle ABC with ∠C90°. Let ∠A30° and (triangle AD equiv l_{angleCAB}.

Therefore, ∠CAD 15°.

Since BCx, AB2x, and using the Pythagorean theorem, we find:

(AC^2 4x^2 - x^2) or (AC sqrt{3}x).

From the triangle similarity, we get:

(frac{DB}{AB} frac{CD}{AC}) or (frac{DB}{CD} frac{AB}{AC} frac{2}{sqrt{3}})

Let (DB 2y) and (CD sqrt{3}y). Solving for (y), we get:

(2ysqrt{3}y x) or (y frac{x}{2sqrt{3}})

Thus, (CD sqrt{3}x/2sqrt{3} frac{2sqrt{3}x - 3x}{2sqrt{6-3sqrt{3}}}).

To find (sin15°), we need to find (AD). Using the values from the previous steps, we get:

(AD frac{sqrt{24-12sqrt{3}}}{x})

Therefore, the final value is:

(sin 15° frac{sqrt{2-sqrt{3}}}{2})

Simplifying this, we get:

(sin 15° frac{sqrt{6} - sqrt{2}}{4})

Conclusion

The value of (sin 15°) is a fundamental concept in trigonometry and has applications in various fields. Understanding the multiple methods to calculate it not only reinforces the principles of trigonometry but also enhances problem-solving skills in mathematical analysis.

By exploring these different methods, we can appreciate the elegance and precision of trigonometric identities and their derivations.