Find The Exact Value Of Sin 105

Finding the Exact Value of sin 105°: A Comprehensive Guide

Determining the exact value of trigonometric functions for angles beyond the standard 0°, 30°, 45°, 60°, and 90° often requires utilizing trigonometric identities and properties. Finding the exact value of sin 105° is a classic example that showcases these techniques. This article provides a detailed walkthrough of several methods, explaining the underlying principles and offering a deeper understanding of trigonometry.

Understanding the Problem: Why Not a Calculator?

While a calculator readily provides an approximate decimal value for sin 105°, finding the exact value necessitates a different approach. Calculators offer approximations due to the inherent limitations of representing irrational numbers (like √3 and √2) in decimal form. The exact value preserves the mathematical precision and elegance of the solution, avoiding rounding errors.

Method 1: Using the Sum-to-Product Formula

This method leverages the sum-to-product formula for sine:

sin(A + B) = sin A cos B + cos A sin B

We can express 105° as the sum of two angles with known trigonometric values: 60° and 45°.

Therefore, we have:

sin 105° = sin (60° + 45°)

Applying the sum formula:

sin 105° = sin 60° cos 45° + cos 60° sin 45°

Substituting the known values:

sin 60° = √3/2 cos 60° = 1/2 sin 45° = √2/2 cos 45° = √2/2

sin 105° = (√3/2)(√2/2) + (1/2)(√2/2)

Simplifying:

sin 105° = (√6 + √2) / 4

This is the exact value of sin 105°.

Method 2: Using the Difference Formula

Alternatively, we can express 105° as the difference of two angles:

105° = 150° - 45° or 135° - 30°

Let's use the difference formula:

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

Using 150° - 45°:

sin 105° = sin (150° - 45°) = sin 150° cos 45° - cos 150° sin 45°

We know:

sin 150° = sin (180° - 30°) = sin 30° = 1/2 cos 150° = cos (180° - 30°) = -cos 30° = -√3/2 sin 45° = √2/2 cos 45° = √2/2

Substituting:

sin 105° = (1/2)(√2/2) - (-√3/2)(√2/2) = (√2 + √6) / 4

This yields the same exact value as Method 1.

Method 3: Using the Half-Angle Formula

The half-angle formula for sine is:

sin(A/2) = ±√[(1 - cos A) / 2]

We can utilize this by considering 105° as half of 210°.

sin 105° = sin (210°/2)

Since 105° is in the second quadrant where sine is positive, we take the positive square root:

sin 105° = √[(1 - cos 210°) / 2]

cos 210° = cos (180° + 30°) = -cos 30° = -√3/2

Substituting:

sin 105° = √[(1 + √3/2) / 2] = √[(2 + √3) / 4] = √(2 + √3) / 2

This expression, while seemingly different, is equivalent to (√6 + √2) / 4. To prove this, we can square both expressions and show they are equal. This algebraic manipulation is shown below:

Proof of Equivalence:

Let's square both expressions:

[(√6 + √2) / 4]² = (6 + 2√12 + 2) / 16 = (8 + 4√3) / 16 = (2 + √3) / 4

[√(2 + √3) / 2]² = (2 + √3) / 4

Since the squares are equal, and both original expressions are positive, the expressions are equivalent.

Method 4: Complex Numbers (Advanced)

A more advanced approach involves using Euler's formula:

e^(ix) = cos x + i sin x

where 'i' is the imaginary unit (√-1).

We can express sin 105° using this formula and then manipulate the resulting complex number to find the exact value. This method is more complex and requires a strong understanding of complex number arithmetic. We won't delve into the details here, as the previous methods provide more straightforward solutions.

Choosing the Best Method

The most efficient method depends on your familiarity with trigonometric identities. Methods 1 and 2 (using the sum or difference formulas) are generally the easiest and most straightforward for this specific problem, given the readily available values for sin 60°, cos 60°, sin 45°, and cos 45°. Method 3 (half-angle formula) requires an extra step of algebraic simplification to prove equivalence but is still a valid approach. Method 4 using complex numbers is generally reserved for more advanced mathematical contexts.

Applications and Further Exploration

Understanding how to derive the exact value of sin 105° isn't merely an academic exercise. This skill is fundamental in various areas, including:

• Calculus: Exact values are crucial for avoiding rounding errors in calculations involving derivatives and integrals of trigonometric functions.

• Physics and Engineering: Precise trigonometric calculations are essential in fields like mechanics, wave theory, and electrical engineering.

• Computer Graphics and Game Development: Accurate trigonometric values are needed for creating realistic animations and simulations.

Further exploration could involve deriving exact values for other angles, practicing with different trigonometric identities, and investigating the relationships between trigonometric functions and complex numbers. Understanding these concepts provides a strong foundation for more advanced mathematical studies.

Conclusion: Mastering Trigonometric Identities

Finding the exact value of sin 105° showcases the power and utility of trigonometric identities. By mastering these identities, you can solve a wide range of trigonometric problems with precision, avoiding the pitfalls of relying solely on approximate calculator values. This knowledge is invaluable for anyone pursuing studies or careers that utilize mathematics, especially trigonometry and calculus. Remember to practice and explore different methods to solidify your understanding. The more you work with these concepts, the more intuitive and efficient your problem-solving skills will become.