Calculating sin105°sin15°: Methods and Simplifications

Understanding and solving trigonometric expressions such as sin105°sin15° is both a fundamental and practical skill in mathematics and engineering. Here, we explore various methods to calculate this product, focusing on trigonometric identities and simplifications.

Introduction to Trigonometric Identities

Trigonometry involves the study of relationships between angles and the lengths of the corresponding sides of triangles. It is a crucial branch of mathematics that is used in various fields such as engineering, physics, and astronomy. One of the fundamental aspects of trigonometry is the use of trigonometric identities, which are expressions that are true for all values of the variables involved. These identities can be used to simplify complex trigonometric expressions.

Method 1: Using Known Values

One straightforward way to find the value of sin105° and sin15° is to use their known values and then multiply them:

sin105° 0.966

cos105° -0.259

Adding the values of sin105° and cos105°:

sin105°   cos105°  0.966   (-0.259)  0.707

This method relies on the direct computation of the values, which may not always be available without a calculator or specific trigonometric tables.

Method 2: Product-to-Sum Formulas

Another method involves using the product-to-sum formulas, which are trigonometric identities that express the product of sines and cosines in terms of sums and differences of angles. The identities we will use here are:

sinAB sinAcosB cosAsinB

sinA - B sinAcosB - cosAsinB

Applying these identities to sin105°sin15°:

sin105°sin15°  sin(60° 45°)sin(60°-45°) (sin60°cos45°   cos60°sin45°)(sin60°cos45° - cos60°sin45°) [ (√3/2) * (1/√2)   (1/2) * (1/√2) ] * [ (√3/2) * (1/√2) - (1/2) * (1/√2) ] [ (√3   1)/(2√2) ] * [ (√3 - 1)/(2√2) ] (1/2√2) * (2) 1/√2

This method provides a more general approach and does not require the specific values of sin105° and cos105°.

Method 3: Using Angle Sum and Difference Identities

Another way to calculate sin105°sin15° is by using angle sum and difference identities. We can express sin105° in terms of sin and cos of 75° and 15°:

sin105°sin15°  sin(75°   30°)sin15° 2sin[(75°   30°)/2]cos[(75° - 30°)/2] 2sin45°cos15° 2 * (1/√2) * (√6   √2)/4 (1/√2) * (3^1/2   1/2) √6/2

This method is particularly useful when dealing with more complex trigonometric expressions involving specific angles.

Conclusion

Calculating the product of sines, such as sin105°sin15°, involves using various trigonometric identities and simplifications. Whether you opt for direct computation, product-to-sum formulas, or angle sum and difference identities, each method offers a unique insight into the problem. Understanding these methods not only helps in solving specific trigonometric problems but also in developing a deeper appreciation for the elegance and utility of trigonometric identities.