Trigonometry Identities: Double Angle, Product-to-Sum, and Sum-to-Product

Trigonometry is a branch of mathematics that studies relationships between angles and sides of triangles. It is an essential part of many fields, including physics, engineering, and architecture. One of the key concepts in trigonometry is the concept of identities. Identities are equations that are true for all values of the variables involved. In this blog post, we will discuss some of the most important trigonometric identities, including the double angle identities, the product-to-sum identities, and the sum-to-product identities.

Double Angle Identities

The double angle identities are a set of identities that relate the trigonometric functions of twice an angle to the trigonometric functions of the angle itself. These identities are very useful in solving trigonometric equations and simplifying trigonometric expressions. The three main double angle identities are:

• Cos 2θ = Cos² θ – Sin² θ
• Sin 2θ = 2 Sin θ Cos θ
• Tan 2θ = 2 Tan θ / (1 – Tan² θ)

Here are some examples of how to use these formulas:

Example 1: Find Cos 60° using the double angle identity for cosine.

We know that Cos 30° = √3/2. Using the double angle identity for cosine, we have:

Cos 60° = Cos (2 * 30°) = Cos² 30° – Sin² 30° = (√3/2)² – (1/2)² = 3/4 – 1/4 = 1/2

Example 2: Find Sin 90° using the double angle identity for sine.

We know that Sin 45° = √2/2 and Cos 45° = √2/2. Using the double angle identity for sine, we have:

Sin 90° = Sin (2 * 45°) = 2 Sin 45° Cos 45° = 2 (√2/2) (√2/2) = 1

Product-to-Sum Identities

The product-to-sum identities are a set of identities that express the product of two trigonometric functions as a sum or difference of trigonometric functions. These identities are useful for simplifying trigonometric expressions and solving trigonometric equations. The main product-to-sum identities are:

• Cos A Cos B = (1/2) [Cos (A – B) + Cos (A + B)]
• Sin A Sin B = (1/2) [Cos (A – B) – Cos (A + B)]
• Cos A Sin B = (1/2) [Sin (A + B) + Sin (A – B)]
• Sin A Cos B = (1/2) [Sin (A + B) – Sin (A – B)]

Example 1: Express Cos 3x Cos x as a sum of trigonometric functions.

Using the product-to-sum identity for Cos A Cos B, we have:

Cos 3x Cos x = (1/2) [Cos (3x – x) + Cos (3x + x)] = (1/2) [Cos 2x + Cos 4x]

Sum-to-Product Identities

The sum-to-product identities are a set of identities that express the sum or difference of two trigonometric functions as a product of trigonometric functions. These identities are useful for simplifying trigonometric expressions and solving trigonometric equations. The main sum-to-product identities are:

• Cos A + Cos B = 2 Cos [(A + B)/2] Cos [(A – B)/2]
• Cos A – Cos B = -2 Sin [(A + B)/2] Sin [(A – B)/2]
• Sin A + Sin B = 2 Sin [(A + B)/2] Cos [(A – B)/2]
• Sin A – Sin B = 2 Cos [(A + B)/2] Sin [(A – B)/2]

Example 1: Express Sin 5x + Sin 3x as a product of trigonometric functions.

Using the sum-to-product identity for Sin A + Sin B, we have:

Sin 5x + Sin 3x = 2 Sin [(5x + 3x)/2] Cos [(5x – 3x)/2] = 2 Sin 4x Cos x

Conclusion

Trigonometric identities are an essential part of trigonometry. They are used to solve trigonometric equations, simplify trigonometric expressions, and prove other trigonometric identities. The double angle identities, the product-to-sum identities, and the sum-to-product identities are some of the most important trigonometric identities. By understanding these identities, you can gain a deeper understanding of trigonometry and its applications.