(a) Use the formula e^2 i θ=(e^i θ)^2 to deduce the well-known trigonometric identities for cos2

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(a) Use the formula e^2 i θ=(e^i θ)^2 to deduce the...

(a) Use the formula $e^{2 \mathrm{i} \theta}=\left(e^{\mathrm{i} \theta}\right)^2$ to deduce the well-known trigonometric identities for $\cos 2 \theta$ and $\sin 2 \theta$. (b) Derive the corresponding identities for $\cos 3 \theta$ and $\sin 3 \theta$. (c) Write down the explicit identities for $\cos m \theta$ and $\sin m \theta$ as polynomials in $\cos \theta$ and $\sin \theta$.

(a) Use the formula $e^{2 \mathrm{i} \theta}=\left(e^{\mathrm{i} \theta}\right)^2$ to deduce the well-known trigonometric identities for $\cos 2 \theta$ and $\sin 2 \theta$. (b) Derive the corresponding identities for $\cos 3 \theta$ and $\sin 3 \theta$.
(c) Write down the explicit identities for $\cos m \theta$ and $\sin m \theta$ as polynomials in $\cos \theta$ and $\sin \theta$.

Key Concepts

Polynomial Expressions of Trigonometric Functions

In the identities for cos m? and sin m?, the functions are expressed as polynomials in cos? and sin?. This polynomial formulation arises from the algebraic expansion provided by De Moivre's theorem and plays a key role in advanced trigonometry, allowing for the analysis and simplification of trigonometric expressions in terms of their fundamental components.

Double-Angle and Triple-Angle Formulas

Double-angle and triple-angle formulas are specific cases of multiple-angle identities derived using Euler's formula or De Moivre's theorem. They express cos2? and sin2?, as well as cos3? and sin3?, in terms of cos? and sin?. These identities are useful in solving equations, integrating trigonometric functions, and simplifying expressions in various areas of mathematics.

De Moivre's Theorem

De Moivre's theorem states that (cos? + i sin?)^m = cos(m?) + i sin(m?) for any integer m. This theorem generalizes Euler's formula to multiple angles and simplifies the derivation of identities for cos m? and sin m? by expanding the power expression and equating the real and imaginary parts.

Euler's Formula

Euler's formula, given by e^(i?) = cos? + i sin?, is fundamental in linking complex exponentials with trigonometric functions. It provides a concise representation of sine and cosine, making it an essential tool for deriving trigonometric identities through algebraic manipulation of exponential expressions.

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