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 employing trigonometric identities and strategic manipulation. This article provides a detailed exploration of calculating the exact value of sin 105°, demonstrating multiple methods and reinforcing fundamental trigonometric concepts.

Understanding the Problem: Why Not Just Use a Calculator?

While a calculator readily provides an approximate decimal value for sin 105°, finding the exact value requires a deeper understanding of trigonometric identities and the unit circle. Approximate values are insufficient for many mathematical applications, particularly those requiring precise calculations or analytical solutions. Therefore, mastering techniques to derive exact trigonometric values is crucial for advanced mathematical study.

Method 1: Using the Sum-to-Product Formula

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

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

We can express 105° as the sum of two angles whose sine and cosine values are known: 60° and 45°. Therefore:

sin 105° = sin (60° + 45°)

Applying the sum-to-product 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

We get:

sin 105° = (√3/2)(√2/2) + (1/2)(√2/2)

Simplifying:

sin 105° = (√6 + √2) / 4

Verifying the Result

While we've derived the exact value, it's beneficial to verify it using a different approach to ensure accuracy. This cross-validation enhances confidence in our solution.

Method 2: Using the Difference Formula

Alternatively, we can express 105° as the difference between two known angles: 150° and 45°. The difference formula for sine is:

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

Therefore:

sin 105° = sin (150° - 45°)

Applying the difference formula:

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

Substituting known values (remembering that sin 150° = sin 30° = 1/2 and cos 150° = -cos 30° = -√3/2):

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

Simplifying:

sin 105° = (√2 + √6) / 4

This result matches the value obtained using the sum-to-product formula, confirming our calculation.

Method 3: Half-Angle Formula

The half-angle formula provides another route to the solution. Since 105° is half of 210°, we can utilize the half-angle formula for sine:

sin(A/2) = ±√[(1 - cos A) / 2]

In this case, A = 210°, so:

sin 105° = ±√[(1 - cos 210°) / 2]

Since 210° lies in the third quadrant where sine is negative, we consider the negative root. cos 210° = -√3/2. Substituting:

sin 105° = -√[(1 + √3/2) / 2]

To simplify this expression, we rationalize the denominator:

sin 105° = -√[(2 + √3) / 4]

This form might seem different, but it's equivalent to (√6 + √2) / 4, as we can prove through further algebraic manipulation. The key is to recognize that squaring the expression -√[(2 + √3) / 4] and simplifying will lead to the same result as squaring (√6 + √2) / 4.

Method 4: Using Complex Numbers (Euler's Formula)

This approach utilizes Euler's formula, which elegantly connects exponential functions and trigonometric functions:

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

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

For sin 105°, we can write:

e^(i(105°)) = cos 105° + i sin 105°

By expressing 105° as 60° + 45°, and then using the properties of exponents and the known values of e^(i60°) and e^(i45°), we can solve for sin 105° through complex number manipulation. This method, while powerful, is more advanced and requires a strong understanding of complex numbers.

Significance and Applications

Knowing the exact value of sin 105° is essential in various mathematical and scientific fields:

• Advanced Trigonometry: Solving complex trigonometric equations and identities often requires working with exact values rather than approximations.

• Calculus: Exact values are crucial for evaluating definite integrals involving trigonometric functions.

• Physics and Engineering: Many physical phenomena are modeled using trigonometric functions. Precise calculations are needed for accurate predictions and designs.

• Computer Graphics: Exact values ensure accurate rendering of shapes and figures in computer graphics and animation.

Conclusion

This article has comprehensively explored multiple methods for finding the exact value of sin 105°. The ability to derive these values is fundamental for anyone seeking to advance their knowledge of trigonometry and its applications. By mastering these techniques, you'll enhance your mathematical skills and gain a deeper understanding of the intricate relationships within the trigonometric world. Remember that while calculators offer convenience, understanding the underlying mathematical principles remains paramount. The various methods presented offer diverse pathways to reach the same accurate solution, highlighting the elegance and interconnectedness of mathematical concepts. The ability to choose and utilize the most appropriate method based on the problem's context demonstrates a refined understanding of the subject matter.