Transformation formulae

In advanced trigonometry, transformation formulas refer to a group of identities that allow expressions involving trigonometric functions to be changed into different forms. These transformations are particularly useful for simplifying trigonometric equations and solving trigonometric problems. The basic transformation formulas in trigonometry include formulas for the sum, difference, double angle, and half angle identities. Here, we will explore each of these in detail, with examples and visual representations to aid understanding.

Sum and difference identities

The sum and difference identities enable us to express trigonometric functions of the sum or difference of angles as the product of trigonometric functions of individual angles. These identities are extremely useful for simplifying complex trigonometric equations.

Sine sum and difference identities

The sine of the sum of two angles can be expressed as:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

The sine of the difference of two angles can be expressed as:

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Cosine sum and difference identities

The cosine of the sum of two angles can be expressed as:

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

The cosine of the difference of two angles can be expressed as:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

Tangent sum and difference identities

The tangent of the sum of two angles can be expressed as:

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

The tangent of the difference of two angles can be expressed as:

tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b))

Double angle identity

Double angle identities are used to express trigonometric functions of double angles in terms of functions of single angles. These identities help in solving equations where the angle is doubled.

Double angle formulas for sine and cosine

For sine, the double angle identity is:

sin(2a) = 2sin(a)cos(a)

For cosine, there are three versions of the double angle identities:

cos(2a) = cos²(a) - sin²(a)

cos(2a) = 2cos²(a) - 1

cos(2a) = 1 - 2sin²(a)

Double angle formula for tangent

For the tangent, the double angle identity is:

tan(2a) = 2tan(a) / (1 - tan²(a))

Half angle identities

Half angle identities are used when solving trigonometric equations where the angle is half. They are derived from double angle identities and are useful for finding the values of trigonometric functions at half angles.

Half angle formulas for sine and cosine

For sine, the half angle identity is:

sin(a/2) = √((1 - cos(a)) / 2)

For cosine, the half angle identity is:

cos(a/2) = √((1 + cos(a)) / 2)

Half angle formula for tangent

For the tangent, the half angle identity is:

tan(a/2) = √((1 - cos(a)) / (1 + cos(a)))

As an alternative:

tan(a/2) = sin(a) / (1 + cos(a))

tan(a/2) = (1 - cos(a)) / sin(a)

Example

Example 1: Simplify sin(75°)

To simplify sin(75°), note that 75° = 45° + 30°. We can use the sine sum identity:

sin(75°) = sin(45° + 30°)

Apply the summation identity:

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

Substitute the known values:

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

Simplification:

sin(75°) = √6/4 + √2/4 = (√6 + √2)/4

Example 2: Verify the identity cos(2a) = 1 - 2sin²(a)

Start with the double angle identity:

cos(2a) = cos²(a) - sin²(a)

Substitute cos²(a) = 1 - sin²(a):

cos(2a) = (1 - sin²(a)) - sin²(a)

Simplification:

cos(2a) = 1 - 2sin²(a)

Example 3: Solve tan(2x) = 1 for 0 ≤ x < π

Using the double angle identity for tangent:

tan(2x) = 2tan(x) / (1 - tan²(x))

Set it equal to 1:

2tan(x) / (1 - tan²(x)) = 1

Cross-multiply to solve for tan(x):

2tan(x) = 1 - tan²(x)

Rearrange the equation:

tan²(x) + 2tan(x) - 1 = 0

Solve this quadratic equation for tan(x):

tan(x) = (-2 ± √(2² + 4 * 1)) / 2

tan(x) = (-2 ± √8) / 2

tan(x) = -1 ± √2

Find the value of x:

We need tan(x) = -1 + √2 or tan(x) = -1 - √2, find the corresponding angle within this range.

Conclusion

Conversion formulas in trigonometry are an essential set of tools for solving and simplifying trigonometric expressions and equations. By converting complex expressions using these identities, we are able to tackle more complex problems in a logical, step-by-step manner. Whether dealing with angle sums, differences, double or half angles, these formulas significantly enhance our ability to work effectively with trigonometric functions.

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