Reducing and enlarging formulas?

Reducing and enlarging formulas involves simplifying or expanding mathematical expressions to make them easier to work with or to better understand their relationships.

To reduce a formula, you can simplify it by combining like terms, factoring out common factors, or using properties of exponents to simplify expressions. This can help make the formula more concise and easier to work with in calculations.

To enlarge a formula, you can expand it by distributing terms, multiplying out parentheses, or using properties of exponents to expand expressions. This can help show the relationships between different variables and make the formula more detailed and explicit.

Overall, reducing and enlarging formulas can help make them more manageable and easier to work with in various mathematical contexts.

Reducing Formulas

Trigonometric Functions:

- sin(α/2) = ±√((1 - cos α) / 2)
- cos(α/2) = ±√((1 + cos α) / 2)
- tan(α/2) = ±√((1 - cos α) / (1 + cos α))

Inverse Trigonometric Functions:

- arcsin(sin(α/2)) = α/2
- arccos(cos(α/2)) = α/2
- arctan(tan(α/2)) = α/2

Exponential and Logarithmic Functions:

- e^(a/2) = ±√(e^a)
- log(a/b) = log(a) - log(b)

Enlarging Formulas

Trigonometric Functions:

- sin 2α = 2 sin α cos α
- cos 2α = cos^2 α - sin^2 α = 2 cos^2 α - 1 = 1 - 2 sin^2 α
- tan 2α = (2 tan α) / (1 - tan^2 α)

Inverse Trigonometric Functions:

- arcsin 2α = arctan √((1 - 2α^2) / 2α^2))
- arccos 2α = arctan √((1 + 2α^2) / 2α^2))
- arctan 2α = arctan (2α / (1 - 2α^2))

Exponential and Logarithmic Functions:

- e^2a = (e^a)^2
- log(a^b) = b log(a)