Examples Of Sum And Difference Identities

Let's delve into the fascinating world of trigonometric identities, specifically focusing on sum and difference identities. These identities are powerful tools that allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. Understanding and applying these identities opens doors to simplifying complex trigonometric expressions, solving equations, and exploring deeper mathematical concepts.

Understanding Sum and Difference Identities

The sum and difference identities are a cornerstone of trigonometry. They provide a way to break down trigonometric functions of combined angles into more manageable components. Before diving into examples, let's lay out the core 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)
• 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)
• 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 are not just abstract formulas; they are derived from fundamental geometric relationships and can be visualized using the unit circle. Mastering these identities is crucial for success in trigonometry and calculus.

Examples with Sine

Let's explore some examples of how to use the sine sum and difference identities.

Example 1: Finding sin(75°)

We can express 75° as the sum of two common angles: 45° + 30°. We know the sine and cosine values for these angles:

• sin(45°) = √2 / 2
• cos(45°) = √2 / 2
• sin(30°) = 1/2
• cos(30°) = √3 / 2

Now, apply the sine of a sum identity:

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

Substitute the known values:

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

Therefore, sin(75°) = (√6 + √2) / 4.

Example 2: Finding sin(15°)

We can express 15° as the difference of two common angles: 45° - 30°. Using the same sine and cosine values as above, apply the sine of a difference identity:

sin(15°) = sin(45° - 30°) = sin(45°)cos(30°) - cos(45°)sin(30°)

Substitute the known values:

sin(15°) = (√2 / 2)(√3 / 2) - (√2 / 2)(1/2) = (√6 - √2) / 4

Therefore, sin(15°) = (√6 - √2) / 4.

Example 3: Simplifying sin(x + π/2)

This example demonstrates the power of these identities for simplifying expressions involving variables. Apply the sine of a sum identity:

sin(x + π/2) = sin(x)cos(π/2) + cos(x)sin(π/2)

We know that cos(π/2) = 0 and sin(π/2) = 1. Substitute these values:

sin(x + π/2) = sin(x)(0) + cos(x)(1) = cos(x)

Therefore, sin(x + π/2) simplifies to cos(x). This result is a fundamental trigonometric relationship.

Example 4: Simplifying sin(x - π)

Applying the sine of a difference identity:

sin(x - π) = sin(x)cos(π) - cos(x)sin(π)

We know that cos(π) = -1 and sin(π) = 0. Substitute these values:

sin(x - π) = sin(x)(-1) - cos(x)(0) = -sin(x)

Therefore, sin(x - π) simplifies to -sin(x).

Example 5: Solving a Trigonometric Equation

Solve the equation: sin(x + π/4) - sin(x - π/4) = 1

Apply the sine of a sum and difference identities:

[sin(x)cos(π/4) + cos(x)sin(π/4)] - [sin(x)cos(π/4) - cos(x)sin(π/4)] = 1

Simplify, knowing that sin(π/4) = cos(π/4) = √2 / 2:

[sin(x)(√2 / 2) + cos(x)(√2 / 2)] - [sin(x)(√2 / 2) - cos(x)(√2 / 2)] = 1

sin(x)(√2 / 2) + cos(x)(√2 / 2) - sin(x)(√2 / 2) + cos(x)(√2 / 2) = 1

2cos(x)(√2 / 2) = 1

cos(x)√2 = 1

cos(x) = 1/√2 = √2 / 2

Therefore, x = π/4 + 2πk or x = 7π/4 + 2πk, where k is an integer.

Examples with Cosine

Now, let's explore examples using the cosine sum and difference identities.

Example 1: Finding cos(75°)

As before, we can express 75° as the sum of two common angles: 45° + 30°. Apply the cosine of a sum identity:

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

Substitute the known values:

cos(75°) = (√2 / 2)(√3 / 2) - (√2 / 2)(1/2) = (√6 - √2) / 4

Therefore, cos(75°) = (√6 - √2) / 4.

Example 2: Finding cos(15°)

Express 15° as the difference of two common angles: 45° - 30°. Apply the cosine of a difference identity:

cos(15°) = cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°)

Substitute the known values:

cos(15°) = (√2 / 2)(√3 / 2) + (√2 / 2)(1/2) = (√6 + √2) / 4

Therefore, cos(15°) = (√6 + √2) / 4.

Example 3: Simplifying cos(x + π/2)

Apply the cosine of a sum identity:

cos(x + π/2) = cos(x)cos(π/2) - sin(x)sin(π/2)

We know that cos(π/2) = 0 and sin(π/2) = 1. Substitute these values:

cos(x + π/2) = cos(x)(0) - sin(x)(1) = -sin(x)

Therefore, cos(x + π/2) simplifies to -sin(x). This is another fundamental trigonometric relationship.

Example 4: Simplifying cos(x - π)

Applying the cosine of a difference identity:

cos(x - π) = cos(x)cos(π) + sin(x)sin(π)

We know that cos(π) = -1 and sin(π) = 0. Substitute these values:

cos(x - π) = cos(x)(-1) + sin(x)(0) = -cos(x)

Therefore, cos(x - π) simplifies to -cos(x).

Example 5: Proving an Identity

Prove the identity: cos(A + B) + cos(A - B) = 2cos(A)cos(B)

Start with the left-hand side (LHS) and apply the cosine sum and difference identities:

LHS = cos(A + B) + cos(A - B) = [cos(A)cos(B) - sin(A)sin(B)] + [cos(A)cos(B) + sin(A)sin(B)]

Simplify:

LHS = cos(A)cos(B) - sin(A)sin(B) + cos(A)cos(B) + sin(A)sin(B) = 2cos(A)cos(B)

Since the LHS simplifies to 2cos(A)cos(B), which is the right-hand side (RHS), the identity is proven.

Examples with Tangent

Let's move on to examples using the tangent sum and difference identities.

Example 1: Finding tan(75°)

Express 75° as the sum of two common angles: 45° + 30°. We know the tangent values for these angles:

• tan(45°) = 1
• tan(30°) = 1/√3 = √3 / 3

Apply the tangent of a sum identity:

tan(75°) = tan(45° + 30°) = (tan(45°) + tan(30°)) / (1 - tan(45°)tan(30°))

Substitute the known values:

tan(75°) = (1 + √3 / 3) / (1 - 1 * √3 / 3) = ( (3 + √3) / 3 ) / ( (3 - √3) / 3 ) = (3 + √3) / (3 - √3)

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator (3 + √3):

tan(75°) = ((3 + √3) / (3 - √3)) * ((3 + √3) / (3 + √3)) = (9 + 6√3 + 3) / (9 - 3) = (12 + 6√3) / 6 = 2 + √3

Therefore, tan(75°) = 2 + √3.

Example 2: Finding tan(15°)

Express 15° as the difference of two common angles: 45° - 30°. Using the same tangent values as above, apply the tangent of a difference identity:

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

Substitute the known values:

tan(15°) = (1 - √3 / 3) / (1 + 1 * √3 / 3) = ( (3 - √3) / 3 ) / ( (3 + √3) / 3 ) = (3 - √3) / (3 + √3)

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator (3 - √3):

tan(15°) = ((3 - √3) / (3 + √3)) * ((3 - √3) / (3 - √3)) = (9 - 6√3 + 3) / (9 - 3) = (12 - 6√3) / 6 = 2 - √3

Therefore, tan(15°) = 2 - √3.

Example 3: Simplifying tan(x + π/4)

Apply the tangent of a sum identity:

tan(x + π/4) = (tan(x) + tan(π/4)) / (1 - tan(x)tan(π/4))

We know that tan(π/4) = 1. Substitute this value:

tan(x + π/4) = (tan(x) + 1) / (1 - tan(x))

Therefore, tan(x + π/4) simplifies to (tan(x) + 1) / (1 - tan(x)).

Example 4: Proving an Identity

Prove the identity: tan(x + y) - tan(x - y) = (2tan(y)(1 + tan²(x))) / (1 - tan²(x)tan²(y))

This proof is a bit more involved. Start with the left-hand side (LHS) and apply the tangent sum and difference identities:

LHS = tan(x + y) - tan(x - y) = [(tan(x) + tan(y)) / (1 - tan(x)tan(y))] - [(tan(x) - tan(y)) / (1 + tan(x)tan(y))]

Find a common denominator:

LHS = [ (tan(x) + tan(y))(1 + tan(x)tan(y)) - (tan(x) - tan(y))(1 - tan(x)tan(y)) ] / [ (1 - tan(x)tan(y))(1 + tan(x)tan(y)) ]

Expand the numerator:

LHS = [ tan(x) + tan(x)²tan(y) + tan(y) + tan(x)tan(y)² - (tan(x) - tan(x)²tan(y) - tan(y) + tan(x)tan(y)²) ] / [ 1 - tan(x)²tan(y)² ]

Simplify the numerator:

LHS = [ 2tan(y) + 2tan(x)²tan(y) ] / [ 1 - tan(x)²tan(y)² ] = [ 2tan(y)(1 + tan(x)²) ] / [ 1 - tan(x)²tan(y)² ]

This matches the right-hand side (RHS) of the identity. Therefore, the identity is proven. Note: There seems to be a typo in the original identity. It should be tan²(x) in the numerator.

Applications of Sum and Difference Identities

Sum and difference identities are not just theoretical tools; they have numerous applications in various fields:

• Physics: Analyzing wave phenomena, such as sound and light, often involves trigonometric functions of sums and differences of angles. These identities help simplify calculations related to interference and superposition.
• Engineering: Signal processing, control systems, and structural analysis frequently utilize trigonometric functions. Sum and difference identities are essential for manipulating and analyzing these functions.
• Computer Graphics: Creating realistic images and animations often involves complex transformations. Trigonometric identities, including sum and difference identities, are used to perform rotations and other geometric operations efficiently.
• Navigation: Calculating distances and bearings in navigation relies on trigonometric principles. Sum and difference identities can be used to simplify calculations involving angles.
• Pure Mathematics: These identities are fundamental building blocks for more advanced topics in trigonometry, calculus, and complex analysis.

Common Mistakes to Avoid

When working with sum and difference identities, it's easy to make mistakes. Here are some common pitfalls to avoid:

• Incorrectly applying the formulas: Double-check that you are using the correct formula for the sine, cosine, or tangent of a sum or difference. Pay close attention to the signs.
• Confusing sine and cosine: Remember that the sine of a sum involves both sine and cosine of the individual angles, while the cosine of a sum involves the product of cosines minus the product of sines.
• Forgetting to distribute: When simplifying expressions, be sure to distribute correctly, especially when dealing with multiple terms.
• Not simplifying completely: Always simplify your answers as much as possible. This may involve combining like terms, rationalizing denominators, or using other trigonometric identities.
• Assuming identities work for all values: While these identities are generally true, be mindful of potential restrictions, such as values that make the denominator of a tangent function equal to zero.

Conclusion

The sum and difference identities are powerful and versatile tools in trigonometry. They allow us to express trigonometric functions of combined angles in terms of trigonometric functions of individual angles, enabling us to simplify expressions, solve equations, and explore deeper mathematical concepts. By understanding and practicing with these identities, you can significantly enhance your problem-solving skills in trigonometry and related fields. The examples provided offer a solid foundation for further exploration and application of these fundamental trigonometric relationships. Remember to practice regularly and pay attention to the details to master these essential tools.