Trigonometry: Sum and Difference Identities

Trigonometry is a branch of mathematics that studies relationships between angles and sides of triangles. It is a fundamental concept in many fields, including physics, engineering, and architecture. One of the key aspects of trigonometry is the understanding of trigonometric identities, which are equations that are true for all values of the variables involved. These identities allow us to simplify expressions, solve equations, and prove other trigonometric relationships.

In this article, we’ll explore a set of important trigonometric identities known as the **sum and difference identities**. These identities relate trigonometric functions of the sum or difference of two angles to trigonometric functions of the individual angles. They are essential tools for solving various problems in trigonometry and related fields.

Sine of a Sum and Difference

The sine of a sum or difference of two angles can be expressed using the following identities:

• **Sine of a Sum:** sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• **Sine of a Difference:** sin(A – B) = sin(A)cos(B) – cos(A)sin(B)

These identities can be derived using the unit circle and the properties of right triangles. The key idea is to express the sine of the sum or difference of two angles in terms of the sine and cosine of the individual angles.

Example:

Let’s say we want to find the value of sin(75°). We can use the sine of a sum identity to express sin(75°) as sin(45° + 30°):

sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°)

We know that sin(45°) = cos(45°) = √2/2, sin(30°) = 1/2, and cos(30°) = √3/2. Substituting these values, we get:

sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

Cosine of a Sum and Difference

The cosine of a sum or difference of two angles can be expressed using the following identities:

• **Cosine of a Sum:** cos(A + B) = cos(A)cos(B) – sin(A)sin(B)
• **Cosine of a Difference:** cos(A – B) = cos(A)cos(B) + sin(A)sin(B)

These identities can be derived using the unit circle and the properties of right triangles, similar to the sine identities.

Tangent of a Sum and Difference

The tangent of a sum or difference of two angles can be expressed using the following identities:

• **Tangent of a Sum:** tan(A + B) = (tan(A) + tan(B)) / (1 – tan(A)tan(B))
• **Tangent of a Difference:** tan(A – B) = (tan(A) – tan(B)) / (1 + tan(A)tan(B))

These identities can be derived using the sine and cosine identities, and the fact that tan(x) = sin(x)/cos(x).

Example:

Let’s say we want to find the value of tan(15°). We can use the tangent of a difference identity to express tan(15°) as tan(45° – 30°):

tan(15°) = tan(45° – 30°) = (tan(45°) – tan(30°)) / (1 + tan(45°)tan(30°))

We know that tan(45°) = 1 and tan(30°) = 1/√3. Substituting these values, we get:

tan(15°) = (1 – 1/√3) / (1 + 1/√3) = (√3 – 1) / (√3 + 1)

We can rationalize the denominator by multiplying both numerator and denominator by (√3 – 1):

tan(15°) = ((√3 – 1)(√3 – 1)) / ((√3 + 1)(√3 – 1)) = (3 – 2√3 + 1) / (3 – 1) = 2 – √3

Applications of Sum and Difference Identities

The sum and difference identities have various applications in trigonometry and related fields. Some of the key applications include:

• **Simplifying trigonometric expressions:** The identities can be used to simplify complex trigonometric expressions by replacing sums or differences of angles with single angle expressions.
• **Solving trigonometric equations:** The identities can be used to solve equations involving trigonometric functions of sums or differences of angles.
• **Proving other trigonometric identities:** The identities can be used as building blocks to prove other trigonometric relationships.
• **Applications in physics and engineering:** The identities are used in various fields, including physics (wave phenomena, optics) and engineering (circuit analysis, signal processing).

Conclusion

The sum and difference identities are fundamental tools in trigonometry. They provide a way to relate trigonometric functions of sums or differences of angles to trigonometric functions of the individual angles. These identities are essential for simplifying expressions, solving equations, proving other identities, and finding solutions in various fields.