Calculating the Value of sin 75° cos 15° Using Trigonometric Identities

Understanding trigonometric identities is a fundamental aspect of mathematics, particularly in solving problems related to angles and their trigonometric functions. In this article, we will explore a specific expression: sin 75° cos 15°, and determine its value using various trigonometric identities. This exercise is not only beneficial for strengthening one's understanding of trigonometry but also serves as a practical application of these identities.

Introduction to Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of their variables. They help simplify and solve complex trigonometric problems. The identities we will use in this article include the product-to-sum identities, co-function identities, and double-angle identities. Understanding these identities is crucial for solving trigonometric expressions such as sin 75° cos 15°.

Using the Co-function Identity

One of the simplest and most straightforward approaches to solving sin 75° cos 15° is to use the co-function identity. According to this identity, sin(90 - θ) cos θ. Therefore, we can rewrite sin 75° as cos 15°, simplifying our expression:

sin75°cos15° cos15°cos15° cos^215°

Next, we can use the double-angle identity for cosine to further simplify cos^2 15°. The double-angle identity for cosine is:

cos^2θ frac{1 cos2θ}{2}

Applying this, we get:

cos^215° frac{1 cos30°}{2} frac{1 frac{sqrt{3}}{2}}{2} frac{2 sqrt{3}}{4}

Using the Product-to-Sum Identity

Another method to solve sin 75° cos 15° involves the product-to-sum identities. These identities allow us to express the product of two trigonometric functions as a sum or difference. The relevant identity for our expression is:

sin A cos B frac{1}{2}[sin(A B) sin(A-B)]

Applying this identity, we get:

sin 75° cos 15° frac{1}{2}[sin(75° 15°) sin(75°-15°)] frac{1}{2}[sin90° sin60°] frac{1}{2}[1 frac{sqrt{3}}{2}] frac{2 sqrt{3}}{4}

Using the Sum and Difference of Angles Formulas

A third approach involves breaking down the angles using basic trigonometric formulas. Recognizing that 75° can be expressed as 45° 30° and 15° as 45° - 30°, we use the sine and cosine sum and difference formulas:

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

Substituting these into our original expression, we get:

sin 75° cos 15° sin(45° 30°) cos 15° - cos(45° 30°) sin 15° (frac{1}{sqrt{2}} cdot frac{sqrt{3}}{2}) cos 15° - (frac{1}{sqrt{2}} cdot frac{1}{2}) sin 15° frac{sqrt{3}}{2sqrt{2}} cos 15° - frac{1}{2sqrt{2}} sin 15° frac{sqrt{3} cos 15° - sin 15°}{2sqrt{2}}

Given that cos 15° and sin 15° can be expressed in terms of cos 45° and sin 45°, we simplify further:

sin 75° cos 15° frac{sqrt{3} cdot frac{sqrt{6} sqrt{2}}{4} - frac{sqrt{6} - sqrt{2}}{4}}{2sqrt{2}} frac{sqrt{18} sqrt{6} - sqrt{6} sqrt{2}}{8sqrt{2}} frac{sqrt{18} sqrt{2}}{8sqrt{2}} frac{3sqrt{2}}{8} frac{3}{8}

Conclusion

In conclusion, the value of sin 75° cos 15° can be determined using various trigonometric identities. Whether through the co-function identity, the product-to-sum identity, or the sum and difference of angles formulas, the result remains the same: sin 75° cos 15° 3/8. Understanding these identities and their applications not only enhances one's problem-solving skills but also deepens the comprehension of trigonometric functions.